BOOSTER - A
2) if cosθ - sinθ =√2 sinθ, prove that sinθ + cosθ = √2 cosθ.
3) If 7 cosθ + 5sinθ =5, find the value of 5 cosθ - 7sinθ ±7
4) If Sec θ + tanθ = x , show that, sin θ = (x²-1)/(x²+1).
5) If sin θ + cosecx =2, show that, sinⁿ θ + cosecsⁿ θ =2.
6) If tan⁴ θ + tan² θ =1, show that, cos⁴ θ + cos² θ =1.
7) If cos⁴x + cos²x =1, show that, tan⁴x + tan²x =1.
8) If sin α , cosα, tanα are in GP, Prove that cot⁶α - cot²α =1.
9) If (secx -1)(sec y - 1)(sec z - 1)= (secx + 1)(sec y + 1)(secz + 1), show that the value of each is ± tan x tan y tanz
10) If (a² - b²)sin θ + 2ab cos θ = a² + b², find the value of tan θ . (a²- b²)/2ab
11) if tan θ + sin θ = m and tan θ - sin θ =n, show that m²- n²= 4√(mn)
12) If x sin³ α + y cos³ α = sinα cosα and xsinα - y cos α =0, then show that, x²+ y² = 1.
10) If x = cosecα - sinα and y= secα - cosα, show that, x²y²(x²+ y²+ 3)= 1.
11) If cosecα + cosec β + cosec γ = 0, then show that
(Sinα + sin β+ sin γ)²= sin²α + sin²β + sin²γ.
12) Find the least value of 9 tan²θ + 4 cot²θ. 12
13) If θ lies in the second quadrant and tan θ= -5/12, find the value of
(2 cosθ)/(1- sinθ). -3
14) If tanθ = 7/15 and sinθ is negative, find the value of
{sin(-θ) + cos(-θ)}/{sec(-θ) + tan(-θ)}. 21/85
15) If α= π/19, show that , (sin23α - sin 3α)/(sin 16α + sin 4α) = - 1.
16) Evaluate {cot(570°) + sin(-330°)}/{tan(-210°) + cosec(-750°)}. -√3/2
17) if A, B, C, D are the angles of a cyclic quadrilateral, Show that,
tanA + tanB + tanC + tanD=0.
18) If n be Integers , show that, sin{nπ + (-1)ⁿ π/4 = 1/√2.
19) if x and y are two distinct real positive numbers , does the relation
sec θ = 2xy/(x²+ y²) hold ? Give reasons . Does not hold
20) If α and β are positive acute angles and cosα = 1/√10 and sinβ = 1/√5, find the value of (α - β). π/4
21) If sinα = 12/13 (0<α < π/2) and cosβ = -3/5 (π< β< 3π/2), find the value of cos(α + β). 33/64
BOOSTER - B
1) If x and y are positive acute angles and cosecx= √5, sec y = √10/3, find the value of cosec(x - y). √50
2) if tan(α+ β) + tan(α - β)= 4 and tan(α+β) tan(α -β) =1 (0<α, β< π/2), find the value of α and β. π/4, π/6
3) Prove that:
{sin(α +β)}/{cos(α - β)} = (tanα + tanβ)/(1+ tanα tanβ).
4) Show that: cosec(x + y)= (cosecx cosec y)/(cot x + cot y).
5) prove that: tan 40° + tan 20°= √3(tan 45° - cot 50° cot 70°).
6) if tan θ + tanφ = x and cot θ + cotφ = y, show that, cot(θ +φ)= 1/x - 1/y.
7) Find the maximum and minimum values of cos θ + √3 sin θ and the corresponding values of θ for which the expression is maximum and minimum. Greatest value=2, when θ=π/3 and least value= -2, when θ= 4π/3
8) if tanα = n/(n +1) and tanβ = 1/(2n +1), show that,
a) α+ β= π/4
b) tan(α + 2β)= 1+ 1/n.
9) if tan β= (sin 2α)/(9+ cos2α), show that, 5 tan(α - β)= 4 tan α.
10) If tan²β = tan(α+ θ) tan(α -θ), then show that,
tan²θ = tan(α+ β) tan(α- β).
11) Prove : cos²(α- β) + cos²β - 2 cos(α- β) cosα cosβ = sin²α.
12) If cos α + cos(α+ β)+ cos(α +β +γ)= 0 and sinα + sin(α +β) + sin(α +β +γ)= 0, show that β = γ = 2π/3.
13) If a sin(θ +α)= b sin(θ+β), show that , tanθ= (b sinβ - a sinα)/(a cosα - b cosβ).
14) Find the simplified value of :
cot(β- γ) cot(γ -α)+ cot(γ - α) cot(α -β)+ cot(α -β) cot(β- γ). 1
15) If {cos(α -β)}/{cos(α + β)} + {cos(γ+ δ)}/(cos(γ -δ)} = 0, show that, tanα tanβ tanγ tanδ = -1
16) If 9x= π, find the value of cosx cos2x cos3x cos4x. 1/16
17) Prove that, 4 cos θ cos(60° - θ) cos(60° + θ)= cos3 θ.
18) Prove that:
sin(π/12) sin(3π/12)sin(5π/12) sin(7π/12)sin(9π/12)sin(11π/12) = 1/32.
19) Prove : tan5° tan55° tan65° tan75°= 1.
20) Prove that: (sin9θ cosθ - cos5θ sin3θ)/(sin16θ sin6θ + cos12θ cos10θ) = tan 6θ.
21) If cos y= m cosx, show that, tan{(x - y)/2} = (m -1)/(m +1) cot{(x + y)/2}.
BOOSTER - C
1) If sinθ = n sin(2α - θ), show that, tan(θ - α)= (n-1)/(n +1) tanα.
2) If (1+ m) sin(θ+α)= (1- m) cos(θ- α), show that tan(π/4 - θ)= m cot(π/4 - α).
3) if sin(3α + θ)= 7 sin(α - θ), show that, tan θ = {sinα(1+ sin²α)}/{cosα(1+ cos²α)}.
4) If tan(α -β)= (sin2β)/{(2n +1) - cos2β}, Show that (tanα)/(tan β) = 1+ 1/n.
5) If cosx + cos y + cos z = 0 and sin x + sin y + sin z =0, then prove that, cos{(x + y)/2} = ± 1/2.
6) If cosα + cosβ = - 27/65, sinα + sinβ = 21/65 and π<α - β < 3π, find the values of sin{(α +β)/2} and cos{(α +β)/2}. sin{(α +β)/2} = 7/√30, cos{(α +β)/2}= 9/√30
7) If (cos α)/a = (cos(α +β))/b = (cos(α +2β))/c = (cos(α +3β))/d,
then show that, b(b + d) = c(c + a).
8) Show that sec2α + tan2α = tan(π/4 + α).
9) Show that (3- 4 cos2α + cos4α)/(3+ 4 cos2α + cos4α) = tan⁴α.
10) Show that: cosec50° + √3 sec50° = 4.
11) Show that: cos(π/7) cos(2π/7)cos(4π/7) = -1/8.
12) If 13θ = π, Show that, cosθ cos2θ cos3θ cos4θ cos5θ cos6θ = 1/2⁶.
13) If (tan3A)/tanA) = k, show that, (sin3A)/(sinA) = 2k/(k -1) and the value of k does not lie between 1/3 and 3.
14) Show that: 16 cos⁵θ = cos5θ + 5 cos3θ + 10 cosθ .
15) Show that: 16 sin⁵θ = sin5θ - 5 sin3θ + 10 sinθ .
16) Show that: sin²x cos⁴x = (1/32) (2+ cos2x - 2 cos4x - cos6x).
17) If α and β are positive acute angles and cos2α = (n cos2β -1)/(n - cos2β), show that, √(n -1) tan α = √(n +1) tanβ, (n > 1).
18) If n tan α = (n +1) tanβ, show that, tan( α -β) = (sin2β)/{(2n +1) - cos2β}.
19) Show that:
(2 cosθ -1)(2 cos2θ -1)(2 cos2²θ -1).....(2 cosⁿ⁻¹θ -1) = (2 cos2ⁿθ +1)/(2 cosθ +1).
20) if cos2β = cos(α +γ) sec(α -γ), show that , tanα, tan β, tan γ are in GP
21) If tan²β = tan(α +β) tan(α -β), show that , sin2β= √2 sinα.
BOOSTER- D
1) If α, β are two different values of θ lying between 0 and 2π which satisfy the equation 6 cosθ + 8 sinθ = 8, find the value of sin(α +β). 24/25
2) Prove that : 4 sin40° - tan40°=√3.
3) Show that the value of (cot 3x)/(cotx) does not lie between 1/3 and 3.
4) Prove that: 2 cosec4θ - sec 2θ = (1- tanθ)cosecθ /(1+ tanθ).
5) Prove that √{(1+ sinθ)/(1- sinθ)} = tan(π/4 + θ/2), (0< θ < π/2)
6) Show that: (sin²24° - sin²6°)(sin²42° - sin²12°)= 1/16.
7) Show that: cos(315/2)°= (-1/2) √{2 + √2}
8) Show that: 2 sin(45/4)°= √[2 - √{2 + √2}].
9) Simplify : sin(144° - x) - sin(144°+ x) + sin(72° - x) - sin(72°+ x). sunx
10) Show that:
sin(β -γ) + sin(γ- α) + sin(α- β) + 4 sin{(β -γ)/2} sin{(γ -α)/2} sin{(α -β)/2}= 0.
11) If 270°< A < 360°, prove that, 2 sin(A/2) = √(1- sinA) - √(1+ sinA).
12) if α and β are two roots of the equation a cos θ + b sin θ = c, show that,
tan{(α +β)/2} = b/a.
If α =β, show that a²+ b²= c²
13) Show that: sin5° - sin67° + sin77° - sin139°+ sin149°= 0.
14) Evaluate: cot(15/2)° - tan(15/2)° - tan(75/2)° + cot(75/2)°. 8
15) If cos θ = (cos u - e)/(1- e cos u), show that, tan(u/2)= √{(1- e)/(1+ e)} tan(θ/2).
16) If A+ B+ C=π, show that
a) sin²A + sin²B - sin²C = 2 sinA sinB sinC.
b) cotB cot C - cot C cot A + cot A cot B = 1.
c) (cotB + cotC)/(tanB + tanC) + (cotC + cotA)/(tanC + tanA) + (cotA + cotB)/(tanA + tanB) = 1.
d) cosA/(sinB sinC) + cosB/(sinC sinA) + cosC/(sinA sin B) = 2.
e) cos(A/2 + cos(B/2) + cos(C/2) = 4 cos{(π - A)/4} cos{(π - B)/4} cos{(π - C)/4}
BOOSTER - E
1) If α+ β+ γ =π, show that,
sin²α + sin²β- sin²γ = 2 sinα sinβ cos γ.
2) If A, B, C are the angles of the triangle ABC, then show that,
cos²(A/2) - sin²(B/2) - sin²(C/2) = 2 sin(A/2 sin(B/2) sin(C/2).
3) If A+ B+ C=π/2 and cotA, cotB , cotC are in AP, show that, cotA cotC = 3.
4) If n be positive integer and sun(π/2n) + cos(π/2n) =√n/2, show that, 4≤ n ≤ 8.
5) Solve:
a) cosmx + cos nx = sin mx + sin nx, m≠ n. (2p+1)π/(m - n), (4p +1)π/2(m + n)
b) sin(3 θ - 30°)= cos(2θ+ 10°), 0°< θ < 180°. 22°, 94°, 130°, 166°
c) sin7θ + sin4θ + sinθ = 0, 0≤ θ≤ π/2. 0,π/4,π/2,2π/9,4π/9
d) cos3θ + cos2θ = sin(3θ/2) + sin(θ/2), 0≤θ≤ 2π. π/7, 5π/7,π, 9π/7, 13π/7
e) sin5x - sin3x - sinx =0, 0°< x < 360°. 15°, 75°,105°,165°,180°,195°,255°,285°,345°
f) cot(θ/2) + cosec(θ/2)= cotθ. 4nπ ± 4π/3
g) 8 cosx cos2x coséx = sin6x/sinx. (2n +1)π/14
h) cotx - 2 sin2x =1. nπ/2+ π/8, nπ - π/4
i) 4 sin2θ cosθ= cosecθ, 0≤ θ≤π. π/8,3π/8, 5π/8, 7π/8
j) sinx + cosx = 1+ sinx cosx. 2nπ, 2nπ + π/2
k) 3 sinx + 4 cosx = 5. 2nπ+ α , where tan α =3/4
l) tan²θ+ sec2θ= 1. nπ, nπ ± π/3
m) eᶜᵒˢ ˣ⁺ ˢᶦⁿˣ ⁻¹ = 1. 2nπ, 2π+ π/2
n) cosx - sinx = cosα + sinα . 2nπ - α , (4n -1)π/2+ α
o) tanx + sec x = 2 cosx, 0≤ x ≤ 2π. π/6, 5π/6
p) tan3α = tanα tan(x - α) tan(x + α). nπ± π/3
p) tan3α = tanα tan(x - α) tan(x + α). nπ± π/3
q) 2 tan2x + tan 3x = tan 5x. nπ, (2n +1)π/16
r) tan(π/4 + θ) - tan(π/4 - θ)= 2 tanθ tan(π/4 - θ) tan(π/4 + θ). nπ
6) Solve for the general values of θ and φ:
cosθ + cosφ = 1; cosθ cosφ = 1/4. θ= 2nπ± π/3, φ = 2mπ±π/3
7) If sec ax + sec bx = 0, show that, the values of x form two arithmetic progressions.
8) If sinθ + sinφ =√3 (cosφ - cosθ), show that, sin3θ + sin3φ= 0.
BOOSTER - F (INVERSE TRIGONOMETRY)
1) Evaluate:
a) sin(2 tan⁻¹(1/5) - tan⁻¹(5/12)). 0
b) tan{sin⁻¹(1/3) + cos⁻¹(1/√3)}. 5/√2
c) sin{2 sin⁻¹(1/√26) + sin⁻¹(12/13)}. 1
d) tan{2 tan⁻¹(1/5) - π/4}. -7/17
2) Prove:
a) 2 tan⁻¹2 + tan⁻¹3 = π+ tan⁻¹(1/3).
b) 2 cos⁻¹(2/√5) + cos⁻¹(3/5) = sin⁻¹(24/25).
c) 4 tan⁻¹(1/5) - tan⁻¹(1/239)= π/4.
d) cos⁻¹(8/17) + cos⁻¹(3/5) + cos⁻¹(36/85) = π
e) cos⁻¹a - sin⁻¹b = cos⁻¹[b √(1 - a²)+ a √(1- b²)].
3) If sin⁻¹x + sin⁻¹y =π/2, show that, 2(x²- xy + y²)= 1+ x⁴+ y⁴.
4) Show that:
a) tan⁻¹(p/q) - tan⁻¹{((p - q)/(p + q)}= π/4.
b) tan⁻¹x + tan⁻¹y + tan⁻¹{(1- x - y - xy)/(1+ x + y - xy)}=π/4
5) If tan⁻¹y= 4 tan⁻¹x, express y as algebraic function of x. (4x - 4x³)/(1- 6x²+ x⁴)
6) If tan⁻¹{a/(b + c)} + tan⁻¹{b/(c + a)}=π/4, show that, a,b,c are three sides of a right d triangle.
7) Solve:
a) tan⁻¹x + tan⁻¹(2x) + tan⁻¹(3x)=π. 1
b) cot⁻¹x + sin⁻¹(1/√5)=π/4. 3
c) tan⁻¹{(x -1)/(x -2)} + tan⁻¹{(x +1)/(x +2)}=π/4. ±1/√2
d) cos⁻¹(8/x) + cos⁻¹(15/x)=π/2. 17
e) sin⁻¹x + sin⁻¹(1 - x) = sin⁻¹√(1- x²). 9,1/2
8) Prove that: tan⁻¹{(2a - b)/b √3} + tan⁻¹{(2b - a)/a √3}=π/3.
9) If α= tan⁻¹{x √3/(2k - x)} and β= tan⁻¹{(2x - k)/k √3} , show that one of the values of
(α- β ) is π/6.
10) Show that: cos{(1/2) cos⁻¹(-1/9)}= 2/3
11) Show that,
a cosθ + b sinθ = √(a²+ b²) cos(θ - tan⁻¹(b/a)) = √(a²+ b²) sin(θ + tan⁻¹(a/b)).
12) Show that:
tan⁻¹{(ap - q)/(aq + p)} + tan⁻¹{(b - a)/(ab +1)} + tan⁻¹{(c - b)/(bc +1)} + tan⁻¹(1/c) = tan⁻¹(p/q).
13) Show that: 2 tan⁻¹a + 2 tan⁻¹b = sin⁻¹[{2(a+ b)(1- ab)}/{(1+ a²)(1+ b²)}].
14) Solve: tan⁻¹{(x +1)/(x -1)} + tan⁻¹{(x -1)/x}= tan⁻¹(7), (x ≠ 0,1). No solution
15) If tan⁻¹(yz/xr) + tan⁻¹(zx/yr) + tan⁻¹(xy/zr)=π/2, show that, x²+ y²+ z²= r².
BOOSTER - G
1) Find the value of
a) cos75°. (√3-1)/2√2
b) sin15°. (√3-1)/2√2
2) If tant= 12/13, find the value of tan(45° + α). 25
3) If tan(α + θ)= 1/2 and tan(α-θ)= 1/3, find the value of tan2θ. 1/7
4) If sin(x - α)= cos(x + α), find the value of tanx. 1
5) The value of √3(tan 55° + tan 95°) - tan55° tan95° is
a) greater than 1. b) less than 1. c) equal to 1. d) none b
6) If sin2θ = 4/5, find the value of sinθ. (0≤θ≤π/4). 1/√5
7) If tan15°= x, show that, x²+ 2√3 x -1=0.
8) If sinθ = (1/2) (a + 1/a), find the value of sin3θ. (-1/2) (a³+ 1/a³)
9) If a≤ sin²x - cos2x ≤ b, find the values of a and b. -1,2
10) Find the greatest value of (cosθ - sinθ)² + cos²(π/4 +θ). 3
11) Find the greatest and least values of cos³θ sinθ - sin³θ cosθ. 1/4,-1/4
12) If 2 cos6θ + 6 cos 2θ +1 = 0, find the principal value of θ. π/3
13) If sinθ = -1/2 and cosθ is positive, find the value of θ in between 0 and 2π. 11π/6
14) Is the equation a cosθ + b sinθ = c, (c²> a²+ b²) solvable? No
15) For what value of p, the equation sinx + p cos x = 2p will be solvable? -1/√3≤ p ≤ 1/√3
16) Find the value of tan⁻¹sin cos⁻¹√(2/3). π/6
17) Find the principal value of sin⁻¹ sin(sin(5π/6))ᶜ. (1/2) Radiant
18) Find the value of cos⁻¹x + cos⁻¹(-x), when 0< x < 1. π
19) Find the value of tan{cos⁻¹(4/5) + tan⁻¹(2/3)}. 17/6
20) Find the value of tan{(1/2) (tan⁻¹x + tan⁻¹(1/x)}. 1
21) If A+ B+ C =π and A= tan⁻¹2, B= tan⁻¹3, show that, C=π/4.
BOOSTER - H
1) The value of sin{sin⁻¹(tan(7π/6))+ cos⁻¹(cos(7π/3))} is
a) 0 b) -1 c) 1 d) none
2) Find the general value of cos⁻¹{(1/2) (-1)ⁿ}. mπ±π/3
3) If tan⁻¹(1/3)= 18°26', find the value of tan⁻¹(1/2). 26°34'
4) If sin⁻¹x + sin⁻¹y =2π/3, find the value of cos⁻¹x + cos⁻¹y.
5) Show that tan{(1/2) cos⁻¹a}= √{(1- a)/(1+ a)}.
6) Show that cos⁻¹√(3/5)= (1/2) cos⁻¹(1/5).
7) Three arcs of length l₁, l₂, l₃ of circles of radius r₁, r₂, r₃ subtends at the centre angles θ₁, θ₂, θ₃ respectively. Show that the arc of length l₁ + l₂ + l₃ of a circle of radius (1/n) (r₁θ₁ + r₂θ₂ + r₃θ₃) subtends an angle n radian at the centre.
8) If uₙ = sinⁿα + cosⁿα, show that, (u₃ - u₅)/u₁ = (u₅ - u₇)/u₃.
9) If cos⁴x/cos²y + sin⁴x/sin²y = 1, show that, cos⁴y/cos²x + sin⁴y/sin²x = 1.
10) If (sin⁴α)/a + (cos⁴α)/b = 1/(a+ b), show that (sin⁸α)/a³ + (cos⁸α)/b³ = 1/(a+ b)³.
11) If sec x sec y + tan x tan y = sec z, then show that,
sec x tan y + tan x sec y = ± tan z.
12) If cosec x = cosec y cosec z + cot y cot z, show that,
cosec y= cosec z cosec x ± cot z cot x.
13) Eliminate x and y from the equations a sin²x + b cos²x = c, b sin² y + a cos² y = d, a tan x = b tan y. 1/a + 1/b = 1/c + 1/d
14) If cosx + cos y + cos z + cos x cos y cos z=0, then show that,
sin⁴x(1+ cos y cos z)²= sin⁴y(1+ cosx cos z)²= sin⁴z(1+ cos x cos y)².
15) If a sinθ + b cosθ = a cosecθ - b secθ = 1, then show that a²+ b²= 1+ b²⁾³ - b⁴⁾³.
16) If 0 < x <π/2 and cosx + cos²x + cos³x = 1, find the least value of tan x. √2
17) If the constant a,b,c,k are real and a tanα + b tanβ + c tan γ = k, then show that the minimum value of tan²α + tan²β + tan²γ is k²/(a²+ b²+ c²).
18) If (x/a) cosθ + (y/b) sinθ = 1 and (ax)/cosθ - (by)/sinθ = a²- b², show that, x²/a² + y²/b²= 1.
19) If (x/a) cosθ + (y/b) sinθ = 1 and x sinθ - y cosθ = √(a² sin²θ+ b² cos²θ), then show that x²/a + y¹/b = a+ b.
20) If m²+ m₁²+ 2mm₁ cosθ = 1, n²+ n₁²+ 2nn₁ cosθ = 1 and mn + m₁n₁ + (mn₁ + m₁n) cosθ = 0, show that, cosec²θ= m²+ n².
21) If a sinθ + b cosθ = a cosecθ + b secθ, then show that the value of each side is
(a²⁾³ - b²⁾³) √(a²⁾³ + b²⁾³).
BOOSTER - I
1) If θ = π/4n, show that sin²θ+ sin²3θ + sin²5θ+ .....to 2n terms= n
2) If (a+ b) tan(θ-φ)= (a - b) tan(θ+ φ) and a cos2φ + b cos 2θ = c, show that
a²- b²+ c²= 2ac cos2φ.
3) Eliminate θ from tan(θ+ α)+ tan(θ+ β)= x and cot(θ+ α)+ cot(θ+ β)= y. (x+ y)² tan²(α- β)= x²y² - 4xy
4) If (a cosα sec β - x)/(a sin(α+ β)) = (y - b sinα secβ)/(b cos(α+ β)) = tanβ, show that x²/a²+ y²/b²= 1.
5) If 2 cosθ = a + 1/a and 2 cosφ = b + 1/b, show that 2 cos(θ- φ)= a/b + b/a.
6) If 0 <θ<π/2 show that, cosθ + 2 sinθ > 1.
7) If (cosx + cos y + cos z)/cos(x + y + z) = (sinx + sin y + sin z)/sin(x + y+ z), then show that each side= cos(y + z) + cos(z + x) + cos(x + y).
8) Show: cot16° cot44° + cot44° cot76° - cot76° cot16° = 3
9) Find the least value of 4³ᶜᵒˢˣ + 4⁴ˢᶦⁿˣ. 1/16
10) If θ> 0, φ > 0 and θ + φ =π/3, find the greatest value of tanθ tanφ. 1/3
11) If the equation x²+ px + q = 0 has two roots tanα and tanβ, show that the value of sun²(α+ β) + p sin(α+ β) cos(α+ β) + q cos²(α+ β) is q.
12) If cotα = (a + a²+ a³)¹⁾², cotβ = (1+ a + a⁻¹)¹⁾² and cotγ = (a⁻¹ + a⁻²+ a⁻³)⁻¹⁾², then show that α+ β = γ.
13) √2 cosA = cosB + cos³B and √2 sinA = sinB - sin³B, then show that,
sin(A - B) = ± 1/3.
14) Show that, for real value of θ, the value of 5 cosθ + 3 cos(θ+ π/3) + 3 lies between -4 and 10.
15) If for all values of x, a ≤ 3 cosx + 5 sin(x + π/6)≤ b, then find the values of a and b. -7,7
16) If tan(θ+φ)= a+ b and tan(θ - φ)= a - b, then show that, a tanθ - b tan φ = a²- b².
17) cosθ = sinβ/sinα, cosφ = sinγ/sinα and cos(θ- φ)= sinβ sinγ, then show that, tan²α = tan²β+ tan²γ.
18) If α, β, γ are positive acute angles, then show that,
sinα + sinβ + sinγ > sin(α+ β+ γ).
19) If (sin(α- β))/sinβ = (sin(α+θ))/sinθ, then show that, cotβ - cotθ = cot(α+ θ) + cot(α- β).
20) If n² sin²(α+ β)= sin²α+ sin²β - 2 sinα sinβ cos(α- β), then show that
tanα = (1+ n)tanβ/(1+ n).
21) If tanθ = n tanφ (n > 0), then show that tan²(θ- φ)≤ (n -1)²/4n.
BOOSTER- J
1) Show that (3+ cos4θ) cos2θ = 4(cos⁸θ - sin⁸θ).
2) If cos(θ- α) = p and sin(θ+β)= q, then show that,
p²+ q² - 2pq sin(α+β)= cos²(α+ β).
3) Show that:
a) tan2x tanx + tan3x tan2x = cotx tan3x - 3.
b) cotθ cot2θ + cot2θ cot3θ +2= cotθ(cotθ - cot3θ).
c) tanθ sec4θ - sec2θ tan4θ = tanθ - tan4θ.
4) If u= (1+ cosθ)(1+ cos2θ) - sinθ sin2θ and v= sinθ(1+ cos2θ) + sin2θ(1+ cosθ), show that u²+ v² = 4(1+ cosθ)(1+ cos2θ).
5) Prove:
a) cos(π/7) cos(3π/7)+ cos(3π/7) cos(5π/7)+ cos(5π/7)cos(π/7) = -1/2
b) cos(π/11) +cos(3π/11) +cos(5π/11) + cos(7π/11)+ cos(9π/11) = 1/2
c) cos²40° cos²80°+ cos²80° cos²20+ cos²20° cos²40°= 9/16.
d) cos⁴(π/9) +cos⁴(2π/9) + cos⁴(4π/9)= 9/8.
e) cos⁸(π/8)+ cos⁸(3π/8)+cos⁸(5π/8)+ cos⁸(7π/8)= 17/16.
6) Eliminate θ from the equations
x= a(cosθ + cos2θ) and y= b (sinθ + sin2θ). (x²/a²+ y²/b²)(x²/a²+ y²/b² -3)= 2x/a
7) Eliminate a tanθ + b cot2θ = c and a cotθ - b tan2θ = c. (0<θ< π/4). (b - a)√b = c √(2a - b) - a √(2a)
8) Eliminate α and β:
cosα + cosβ = a, cos2α + cos2β = b and cos3α + cos3β = c. 3a(b +1)= 2a³+ c.
9) Prove: cot(α/2) - 3 cot(3α/2)= (4sinα)/(1+ 2 cosα).
10) If A, B, C are the angles of a triangle and cotθ = cotA + cotB + cotC show that, sin³θ= sin(A - θ) sin(B - θ) sin(C - θ).
11) If cos(A+ B+C)= cosA cosB cosC, show that,
8 sin(B + C) sin(C + A) sin(A+ B)+ sin2A sin2B sin2C= 0.
12) Show that for all real values of θ, cosθ + 3 cos3θ + 6 cos6θ ≥ 115/16.
13) Show that, sinα sin2α sin3α ≤ 9/16. (0<α<π/2)
14) If sinθ + cosθ + tanθ + cotθ + secθ + cosecθ =7, find the value of sin2θ. 22- 8√7
15) If x sinθ + y cosθ = 2α sin2θ and x cosθ - y sinθ = α cos2θ, then show that
(x + y)²⁾³ + (x - y)²⁾³ = 2α²⁾³.
16) If sec2β = 2 secβ cosecβ, show that, cosec2β = cosec²β - sec²β.
17) If tanα tanβ = √{p - q)/p+ q), show that, (p - q cos2α)(p - q cos2β)= p²- q².
18) If (sin²(α+γ))/sin²(β+γ))= (sin2α)/(sin2β), show that, tanα tanβ = tan²γ.
19) If sin2α = (sin2β + sin2γ)/(1+ sin2β sin2γ), show that, tan(π/4+ α) = tan(π/4+ β) tan(π/4 + γ).
20) If (sin(x - θ))/(sin(x - φ))= a/b and (cos(x - θ))/(cos(x - φ))= p/q, show that,
cos(θ- φ)= (ap + bq)/(aq+ bp).
21) If A+ B + C= π and cosA = cot y cot z , cosB= cot z cot x, cosC= cot xcot y, show that cos²x = cotB cotC, cos²y = cotC cotA, cos²z = cotA cotB.
BOOSTER - K
1) If α, β are positive acute angles and 3 sin²α + 2 sin²β = 1 and 3 sin2α - 2 sin2β =0, show that, α + 2β =π/2.
2) If cos α + cosβ + cosγ =0 and sinα + sinβ + sinγ =0, show that,
a) cos²α + cos²β + cos²γ = 3/2
b) sin²α + sin²β+ sin²γ = 3/2.
3) Show that the equation sin x (sin x + cos x)= a has real solution if and only if a is a real number lying between (1/2) (1- √2) and (1/2) (1+ √2).
4) Show that (3+ cos x) cosecx has no value in between -2√2 and+ 2√2.
5) Show that, cosθ(sinθ ± √(sin²θ+ sin²α)) always lies between the values ± √(1+ sin²α).
6) If α, β are the roots of the equation a cos2θ + b sin2θ = c, then show that,
a) tanα + tanβ = 2b/(c + a)
b) tanα tanβ = (c - a)/(c + a).
7) If the equation a cosθ + b sinθ = c is satisfied α and β, two different values of θ, then show that cotα + cotβ = 2ab/(c²- a²).
8) Show: sin(2π/7)+ sin(4π/7)+ sin(8π/7)= √7/2.
9) If α = 2π/7, show that, sinα sin2α sin4α = -√7/8.
10) If α = 2π/7 show that,
tanα tan2α + tan2α tan4α + tan4α tanα = -7.
11) If α = 2π/7, show that, secα + sec2α + sec4α = - 4.
12) If cos³θ/(cos(α- 3θ)) = sin³θ/(sin(α- 3θ)) = k, show that, 2k²- k cosα -1=0.
13) Show: cosec²(π/5) + cosec²(2π/5)= 4.
14) Find the value of: tan²(π/16)+ tan²(3π/16) + tan²(5π/16) + tan²(7π/16). 28
15) If sinα + sinβ = 1/2 and cosα + cosβ = 5/4, then find the value of tanα + tanβ.
16) If sin x + cos y = 1/2 and cos x + sin y= 1/3, then show that, cos(x - y)= 12/13.
17) Eliminate θ and φ from sinθ + sinφ = a, cosθ + cosφ = b and tanθ + tanφ = c. 8ab = c{(a²+ b²)² - 4a²}
18) If cosθ + cosφ = a and sinθ + sinφ = b, show that,
sin2θ + sin2φ = 2ab{1 - 2/(a²+ b²)}.
19) If α and β are the roots of the equation a tanθ + b secθ = c, then show that,
tan(α+ β)= 2ac/(a²- c²).
20) Show: cosA/(1- sinA) + cosB/(1- sinB) = (2sinA - 2 sinB)/{sin(A- B) + cosA - cosB}.
21) If cosx = tan y, cos y= tan z, cos z = tan x, then show that, sinx= sin y = sin z = 2 sin 18°.
BOOSTER - L
1) If tanθ = (sinα cosβ)/(cosα + sinβ), then show that one of the value of tan(θ/2) is tan(α/2) tan(π/4 - β/2).
2) If tan²θ= sinα sinβ sec²{(α+β)/2}, show that one value of tan²(θ/2) is
tan(α/2) tan(β/2).
3) If tanθ = √{(a - b)/(a+ b)} tan(A/2) and cosφ = (b + a cosA)/(a + b cosA), show that,
φ = 2θ.
4) Show that tan18° is one root of the equation 5x⁴- 10x²+ 1=0.
5) If cos²θ= (a²-1)/3 and tan²(θ/2)= tan²⁾³α, then show that, cos²⁾³α + sinα²⁾³α = (2/a)²⁾³.
6) If secα cosθ = secβ cosφ = cosγ and tan(θ/2) tan(φ/2)= tan(γ/2), then show that, sin²γ = (secα -1)(secβ -1).
7) If cosα = cosβ cosφ = cosβ' cosφ' and sinα = 2 sin(φ/2) sin(φ'/2) then show that, tan²(α/2)= tan(β/2) tan²(β'/2).
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8) If tanθ = a/b, show that, a cosec (θ/3) - b sec(θ/3) = 2√(a²+ b²).
9) If tan(π/4 + y/2)= tan³(π/4 + x/2), show that, sin y= (3 sin x + sin³x)/(1+ 3 sin²x).
10) Show: (cos3φ + cos3θ)/{2 cos(θ - φ) -1} =
(cosθ + cosφ) Cos(θ+ φ) - (sinθ + sinφ) sin(θ+ φ).
11) If cosθ/cosα + sinθ/sinα = 1 = cosφ/cosα + sinφ/sin α, then show that,
(cosθ cosφ)/Cos²α + (sinθ sinφ)/sin²α + 1= 0.
12) If α + β+ γ= 2θ, show that, cos²θ+ cos²(θ - α) + cos²(θ - β) + cos²(θ - γ)= 2(1+ cosα cosβ cosγ).
13) If α + β +γ =π/2, show that,
Cos²(π/4) + cos²(π/4 - α) + cos²(π/4 - β) + cos²(π/4 - γ) = 2(1+ cosα cosβ cos γ).
14) If A+ B+ C=π, then show that cotA + sinA/(sinB sinC) retains the same value if any two of the angles A,B, C be interchanged.
15) If x+ y+ z = xyz, then show that:
2x/(1- x²) + 2y/(1- y²) + 2z/(1- z²)= 8xyz/{(1- x²)(1- y²)(1- z²)}.
16) Solve:
a) 8 cos⁶x = 4 + cos2x + 3 cos4x. nπ, (2n + 1)π/4
b) sinx - 3 sin2x + sin3x = cosx - 3 cos2x + cos3x. nπ/2 + π/8
c) sinx + sin2x + sin3x = 1+ cosx + cos2x. (2n +1) π/2, 2nx ±2π/3, nπ +(-1ⁿ), π/6
d) 2(cosx + cos2x) + (1+ 2 cosx) sin2x = 2 sinx, (-π≤ x≤π). -π, -π, -π/3, π/3, π
e) 3 - 2 cosx - 4 sinx - cos2x - sin2x =0. 2nπ, 2nπ + π/2, nπ +(-1ⁿ)π/2
BOOSTER - M
1) Solve:
a) 8(cos2x - cos⁴x) cot3x + (sin5x - sinx) cosec3x = 0. nπ/3 +π/6, nπ + π/4
b) sin³(x - 2π/3) + sin³x + sin³(x + 2π/3)= 0. nπ/3
c) 2+ 2 cos2x cos5x = sin²2x. (2n +1)π,
d) 4 cosx cos2x cos3x - 2 cos4x = 1. (2n +1)π/8, nπ± π/6
e) tan(x + a) tan(x + b)+ (tan(x + b) tan(x + c)+ tan(x + c) tan(x + a)= 1. +2n +1)π/6 - (a+ b+ c)/3
f) tan²x tan²3x tan4x = tan²x - tan²3x + tan 4x. nπ/4, nπ/2 + π/8
g) √(cos²x + 1/2) + √(sin²x + 1/2) = 2. (2n +1)π/4
h) 1+ sin²x = 3 sinx cosx, given tan 18°26'= 1/3. n.180°+ 45°, n 180°+ 26°34'
i) √(tanx + sinx) + √(tanx - sinx)= 2 cosx √tanx. nπ, 4nπ - π/2, 4nπ/3 + π/6
j) tan(x + π/6) tan(x + π/4) + tan(x + π/4) tan(x + π/3) + tan(x + π/3) tan(x + π/6) = 1. nπ/3 - π/12
k) sin⁸x + cos⁸x = (17/16) cos²2x. nπ/4+ π/8
l) sin⁴x + cos⁴x = sinx cosx. nπ+ π/4
m) 2 cosx - cos5x = 16 cos³x sin²x. (2n +1)π/6
n) tanx + tan(x + π/3) + tan(x + 2π/3)= 3. nπ/3+ π/11
o) ₁₆sin²x + ₁₆cos²x = ₁₀, (0≤ x ≤ π). π/6, π/3,2π/3,5π/6
p) ₃(sin2x + 2cos²x) ₊ ₃(1- sin2x + 2 sin²x)₌ ₂₈. nπ- π/4, nπ+ π/2
2) Find the least positive value of x:
tan(x + 100) = tanx tan(x + 50) tan(x - 50). 30°
BOOSTER- N
1) Show that the equation of the equation cos2x + a sinx = 2a - 7 is possible, if 2≤ a≤ 6.
2) Find the range of y such that the equation y+ cosx = sinx has a real solution. If y= 1, find the value of x where 0< x < 2π. -√2≤ y≤√2
3) If x and y are real quantities, show that x + y =2π/3 and cosx + cos y = 3/2 has no solution.
4) θ₁ and θ₂ are two distinct values of θ(0≤ θ₁< 2π, 0≤ θ₂< 2π) satisfying the equation sin( θ+ α) = (1/2) sin2α . Show that, (sinθ₁ + sinθ₂)/(cosθ₁ + cosθ₂)= cot α.
5) If n be an integer and sun(π cot x)= (π tanx), then show that the value of cosec 2x or cot 2x will be in the form of (n + 1/4).
6) Show that the equation sec θ + cosec θ = c has two roots in between 0 and 2π if c²< 8.
7) Show: 2 cos⁻¹(3/√13) + cot⁻¹(16/63) + (1/2 cos⁻¹(7/25)=π.
8) Solve:
a) sec⁻¹(x/a)+ sec⁻¹(x/b) = sec⁻¹(a) + sec⁻¹(b). ab
b) cos⁻¹{(x²-1)/(x²+ 1)} + tan⁻¹{2x)/(x²- 1)}= 2π/3. 1/2,1
c) tan⁻¹{(x-1)/(x+ 1)} + tan⁻¹{(2x-1)/(2x+ 1)} = tan⁻¹{7/6}. 2
d) sun[2cos⁻¹cot(2 tan⁻¹x)]=0. ±1, 1±√2, -1±√2
9) show: cos⁻¹√(2/3) - cos⁻¹{(√6 +1)/(2√3)}= π/6.
10) show: [√50 - √18 - 1/√{3- 2√2}=π/4
11) tan⁻¹{(1/2) tan2A)} + tan⁻¹(cotA) + tan⁻¹(cot⅗A)=π, where 0≤ A≤π/4.
12) Find the value of: cos⁻¹{(x/2) + (1/2) √(3- 3x²)}, 1/2≤ x ≤ 1.
13) If cos⁻¹(x/2) + cos⁻¹(y/3)=0, show that, 9x² - 12xy cosθ + 4y²= 36 sin²θ.
14) Show that the least value of (sin⁻¹x)³ + (cos⁻¹x)³ is π³/32.
15) Show: tan⁻¹√(xr/yz) + tan⁻¹√(yr/zx) + tan⁻¹√(zr/xy) =π, where x+ y + z = r.
16) Show that, 2 tan⁻¹{tan(π/4 - α) tan(β/2)} = cos⁻¹{(sin2α + cosβ)/(1+ sin2α cosβ)}.
17) Show that, 2 tan⁻¹{tan(α/2) tan(π/4 - β/2)} = tan⁻¹{(sinα cosβ)/(cosα + sinβ)}.
18) If tan⁻¹x + tan⁻¹y + tan⁻¹z =π/2 and x+ y + z =√3, show that, x= y = z.
BOOSTER - O
1) Show cos⁻¹x + cos⁻¹y + cos⁻¹z =π and x+ y+ z =3/2, show that, sin⁻¹{2xy/(x²+ y²)}.
2) Express tan⁻¹{2ab/(a² - b²)} + tan⁻¹{2cd/(c² - d²)} in the form tan⁻¹{2xy/(x²- y²)}.
3) If sin⁻¹(x/a) + sin⁻¹(y/b) = sin⁻¹(c²/ab), then show that,
b²x²+ 2xy √(a²b²- c⁴) + a²y²= c⁴.
4) If aα+ sec(tan⁻¹α)= c and αβ + b sec(tan⁻¹β) = c, show that, (α+β)/(1- αβ) = 2ac/(a²- c²).
5) If ψ = sin⁻¹(sinφ + sinθ) - sin⁻¹(sinφ - sinθ) and sin²θ + sin²φ = (1/2) (0<θ<φ<π/2), then show that, cosψ = cos2θ - cos2φ.
6) Show: tan⁻¹ 1+ tan⁻¹[tan²(α+β) tan²(α- β)] = tan⁻¹[(1/2) (cos2α sec2β + cos2β sec2α)].
7) If tan⁻¹√{(a²- x²)/(a²+ x²)} + tan⁻¹√{(b²- y²)/(b²+ y²)}= α/2, then show that, x⁴/a⁴ - (2x²y²) /(a²b²) cosα + y⁴/b⁴ = sin²α.
8) Show: (a³/2) cosec²{(1/2) tan⁻¹(a/b)}+ (b³/2) sec²{(1/2) tan⁻¹(b/a)} = (a + b)(a²+ b²).
9) If {m tan(α-θ)}/cos²θ = {n tanθ)/cos²(α-θ), then show that, θ = (1/2)[α - tan⁻¹{(n - m)/(n + m)} tanα].
10) Show: tan⁻¹{(3 sin2θ)/(5+ 3 cos2θ)} + tan⁻¹{(1/4) tanθ}= θ.
11) If θ = tan⁻¹(2 tan²θ) - (1/2) sin⁻¹{(3 sin2θ)/(5+ 4 cos2θ)} and θ ≠ nπ, show that tanθ is a root of the equation x³- 3x +2=0.
12) Show that tan⁻¹{(2 sin2α)/(1+ 2 cos2α)} - (1/2) sin⁻¹{(3 sin2α)/(5+ 4 cos2α) equals α.
13) Find the simplest value of the following expression:
tan⁻¹[{(1+ a)²p - AP(1+ a²p²)}/{1- a²p² - a(a²+ 1)p²}] + tan⁻¹{(1- a²p²)/(a²+1)p} + tan⁻¹{(1- √3 AP)/(√3 + AP)}.
14) If α, β are obtuse angles and sinα = 1/√10 and sin β = 1/√5, find the value of (α +β). 315°
15) If tan 71° 34' = 3, find the acute angle whose tangent is 1/2. 26°34'
16) Find the least value of 5+ 6 cos θ + 2 cos2 θ. 3/4
17) If cos⁶α + sin⁶α + k sin²2α =1, find the value of k. 3/4
18) If tan9°= a and tan10°= b, find the value of tan2°. {2(b - a)(1+ ab)}/{(1+ ab + b - a)(1+ ab - b + a)}
19) Show that the value of (tanx + 2 tan2x)/tanx can not lie between 1 and 5.
20) If 180°<θ <270 ° and cosθ = -3/5, find the value of tan(θ /4). (√5+1)/2
21) Show that the equation eˢᶦⁿˣ - e⁻ˢᶦⁿˣ = 4 has no real solution.
22) Solve: 2 cos(x/3)= 2ˣ + 2⁻ˣ. 0
23) Is there any solution of ₂sin²x = sinx ?
24) Is the inequality ₃cos² θ + ₃sin² θ ≥ 2√3 true for all values of θ ? Yes
25) If x + 1/x = 2 sin θ, where x is real, find the values of sin θ. ±1
26) The equation sin⁶x + cos⁶x = a is solvable, if
a) 1/2≤ a ≤ 1. b) 1/4≤ a≤ 1 c) -1≤ a≤1. d) 0≤ a≤ 1/2
27) Solve: cosec⁻¹x = cosec⁻¹a + cosec⁻¹b. ab/[√(a²-1) + √(b²-1]
28) Express tan⁻¹a as the sum of two angles of which α is one. tan⁻¹a = α + tan⁻¹{α - tan α)/(1+ a tan⁻¹α)}
29) If x is real and (8x³+ 1)⅖+ (2x +1)²= 0, find the principal value of cos⁻¹x. 2π/3
30) ₑsin⁻¹x + ₑcos⁻¹x. 2e^π/4
31) The number of real solutions of tan⁻¹ √(x²+ x) + sin⁻¹√(x²+ x +1)=π/2.
a) 0 b) 1 c) 2 d) infinite
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