22 a. For acute angles A and B, if sec (2 A – B) = √2 and cosec (A + B) = 2 then find the values of A and B.

22 b. Evaluate: (2 cos 30 – cot ^3 60)/tan 30

31. Prove that: (1 + cot θ – cosec θ) (1 + tan θ + sec θ) = 2

25 a. Prove that : √[(1 + sin A)/(1−sin A)] = sec A + tan A.

25 b. Evaluate : (3 cos2 30◦−6 csc2 30◦)/tan2 60◦

30. Prove that: 1 /(sec x − tan x) − 1/cos x = 1/cos x – 1 / (sec x + tan x)

29 a. If sin θ + cos θ = √3, then prove that tan θ + cot θ = 1

29 b. Prove that: (sin A + sec A) ^ 2 + (cos A + cosec A) ^ 2 = (1 + sec A cosec A) ^ 2

22 a. If tan θ =24/7, then find the value of sin θ + cos θ.

22 b. If cot θ =7/8, then find the value of (1+sin θ)(1−sin θ)/(1+cos θ)(1−cos θ).

26 a. If x = h + a cos θ, y = k + b sin θ, then prove that : ((x−h)/a)^2+ ((y−k)/b)^2 = 1.

26 b. Prove that : tan A/(1+sec A) −tan A/(1−sec A) = 2 cosec A.

25 b. Express cos A and tan A in terms of sin A.

25 a. If tan A = 4/3, find sin A and cos A.

24 b. Prove that: √(1 – sin θ)/(1 + sin θ) = sec θ – tan θ.

24 a. If tan θ + 1/tan θ = 2, find the value of tan^2 θ + 1/tan^2 θ

22 b. Prove that: 1 + (cot^2 α)/(1 + cosec α) = cosec α

22 a. Evaluate: (5cos^2 60 + 4 sec ^ 2 30 – tan ^2 45) / (sin^2 30 + cos^2 30)

26 a. Prove that: sec^3 θ / (sec^2 θ – 1) + cosec^3 θ / (cosec^2 θ – 1) = sec θ . cosec θ (sec θ + cosec θ)

26 b. If sec α / cosec β = p and tan α /cosec β = q, then prove that (p^2 – q^2) * sec^2 α = p^2

If 𝑠𝑖𝑛 (𝐴 + 𝐵) = 1 and 𝑐𝑜𝑠 (𝐴 − 𝐵) = √3/ 2 ,0° < 𝐴, 𝐵 < 90°, find the measure of angles 𝐴 and 𝐵.

Prove that (cos A – sin A + 1) / (cos A + sin A – 1) = cosec A + cot A

If sin θ + cos θ = √3, then prove that tan θ + cot θ = 1

Let 2A + B and A + 2B be acute angles such that sin(2A + B) = (sqrt(3))/2 and tan(A + 2B) = 1 Find the value of cot(4A – 7B)

Prove the following trigonometric identity: (1 + cosec A)/(cosec A) = (cos^2 A)/(1 – sin A)

If sin A = y, then express cos A and tan A in terms of y.

It is given that sin (A – B) = sin A cos B – cos A sin B. Use it to find the value of sin 15°.

If sin A = 3/5, find the values of cos A and tan A

Prove that: sin θ/(cot θ + cosec θ) = 2 + sin θ /(cot θ – cosec θ)

If a cosA + b sinA = m and a sinA – b cosA= n, then a²+b²=

If in a triangle ABC right angled at B, AB = 6 units and BC = 8 units, then find the value of sin A cos C + cos A sin C.

Prove that: (1 + cot² θ) (1 – cos θ) (1 + cos θ) = 1

Prove that (1 + cot^2 A) / (1 + tan^2 A) = cot^2 A

Prove that 1 + tan^2 A = sec^2 A

If sin θ + cos θ = p and sec θ + cosec θ = q, then prove that q (p2 – 1) = 2p

Evaluate: (5 cos 2 60 + 4 sec 2 30 – tan 2 45) / (sin 2 30 + sin 2 60)

If sin (A – B) = 1/2, cos (A + B) = 1/2; 0 < A + B <= 90°, A > B; find ∠ A and ∠ B.

Prove that : (sin θ – cos θ + 1) / (sin θ + cos θ – 1) = 1/ (sec θ – tan θ)

Evaluate : 5 tan 60° / (sin2 60° + cos2 60°) tan 30°

In Δ ABC, if AD ⊥ BC and AD2 = BD x DC, then prove that ∠BAC = 90°

Prove that : (cosec θ – sin θ) (sec θ – cos θ) (tan θ + cot θ) = 1

Evaluate : (sec2 45° – tan2 45°)/sin2 45°

Prove that (sin A + cos A)/(sin A – cos A) + (sin A – cos A)/(sin A + cos A) = 2/(2 sin2 A – 1)

Evaluate: (2𝑠𝑖𝑛 2 60 − 𝑡𝑎𝑛 2 30) / 𝑠𝑒𝑐2 45

If cosθ + sinθ = 1 , then prove that cosθ – sinθ = ±1

Evaluate : (cos 45° + sin 60°) / (sec 30° + cosec 30°).

prove that sin 6 θ + cos 6 θ = 1 – 3 sin 2 θ cos 2 θ

If 2sin(A + B) = √3 and cos(A – B) = 1 , then find the measures of angles A and B. 0 <= A B, (A + B) <= 90 deg

Evaluate: 2 sin 2 30 sec 60 + tan 2 60

Prove that : 1 + sec θ – tan θ / 1 + sec θ + tan θ = 1 – sin θ / cos θ

Evaluate: cos 45 / (sec 30 + cosec 30)

Prove that Root ( sec^2 θ + csc^2 θ) = tan θ + cot θ

If x sin^3 θ + y cos^3 θ = sin θ cos θ and x sin θ = y cos θ, prove that x^2 + y^2 = 1

Evaluate: 2√2 cos 45° sin 30° + 2√3 cos 30°

If A = 60° and B = 30°, verify that : sin (A + B) = sin A cos B + cos A sin B

Prove that : tan θ /(1 – cot θ) + cot θ / (1 – tan θ) = 1 + sec θ cosec θ

If sin A = 3/5 and cos B = 12/13 , then find the value of (tan A + tan B)

If tan θ + sec θ = m, then prove that sec θ = (m2 + 1)/2m

Prove that : tan θ /(1 – cot θ) + cot θ / (1 – tan θ) = 1 + tan θ + cot θ

If x = a sec(theta) + b tan theta and y = a tan theta + b sec(theta), prove that x ^ 2 – y ^ 2 = a ^ 2 – b ^ 2

If sec α = 2/√3, then find the value of  1- cosec α /1+ cosec α   , where α is in IV quadrant.

If cotθ = 15/8, then evaluate (2 + 2 sinθ)(1 – sinθ)/(1 + cosθ)(2 – 2 cosθ)

IF Sec θ=(x+1/4x) then prove that secθ +tanθ =2x or 1/2x

If sec θ + tan θ = p, obtain the values of sec θ, tan θ and sin θ in terms of p.

If sin θ = 3/4, prove that Root [(cosec 2θ − cot 2 θ} / (sec 2 θ − 1)] = √7 / 3

If cosec2θ (1 + cosθ)(1 – cosθ) = λ, then find the value of λ.

Prove that tan A / (1+ sec A) – tan A / (1- sec A) = 2 cosec A

If 1 + sin^2 θ = 3 sin θ cos θ, then prove that tan θ = 1 or 1/2

Find the value of x if 2 cosec2 30 + x sin2 60 – 3/4 tan2 30 = 10

If tan (A + B) = √3 and tan (A – B) = 1/(√3) ; 0° < A + B < 90°; A > B, find A and B.

Prove that: tan θ /(1 – cot θ) + cot θ / (1 – tan θ) = 1 + sec θ cosecθ

Prove that root[(sec A – 1)/(sec A + 1)] + root[(sec A + 1)/(sec A – 1)] = 2 cosec A

If a cos θ + b sin θ = m and a sin θ – b cos θ = n, then prove that a2 + b2 = m2 + n2

If sin θ – cos θ = 0, then find the value of sin^4 θ + cos^4 θ.

Evaluate 2sec^2 θ + 3 cosec^2 θ – 2 sin θ cos θ if θ = 45.

If θ is an acute angle and sin θ = cos θ, find the value of tan^2 θ + cot^2 θ – 2.

Evaluate (5/cot^2 30) + (1/ sin^2 60) – cot^2 45 + 2 sin^2 90

Prove that sec A (1- sin A)(sec A + tan A) = 1

Prove that: (sin A – 2 sin^3 A)/(2cos^3 A – cos A) = tan A

If A and B are acute angles such that sin(A-B) = 0 and 2 cos (A+B) -1 = 0, then find angles A and B.

Evaluate (5cos^2 60 + 4 sec^2 30 – tan ^2 45) / (sin^2 30 + cos^2 30) = 1 + sin θ cos θ

Prove that: cos^2 θ/(1 – tan θ) + sin^3 θ/(sin θ – cos θ) = 1 + sin θ cos θ

Prove that: sin θ / (1 + cos θ) + (1 + cos θ) / sin θ = 2 cosec θ

Prove that: (1/cosθ – cos θ) (1/(sin θ) – sin θ) = 1/(tan θ+ cot θ)

If Cos A + Cos^2 A = 1, then find the value of Sin^2 A + Sin^4 A.

If 4 cot^2 45° — sec2^ 60° + sin^2 60° + p = 3/4 then find the value of p.

Prove that, 2(sin^6 θ + cos^6 θ) – 3 (sin^4 θ + cos^4 θ) +1 = 0

If tan theta = 1/root7, then show that = (cosec sq theta – sec sq theta)/(cosec sq theta + sec sq theta) = 3/4.

If sin θ + sin^2 θ  = 1, then prove that cos^2 θ + cos^4 θ = 1.

Prove that (cosecA – sinA) (secA-cosA) = 1/(cotA+tanA)

Prove that, (tanθ + secθ-1)/(tanθ – secθ + 1) = (1 + sinθ)/cosθ

If Sin α = 1/√2 and Cot β = √3, then find the value of Cosec α + Cosec β.

If Sin θ + Cos θ = √3, then find the value of Sin θ . Cos θ