If y = cot⁻¹(√(cos x)) − tan⁻¹(√(cos x)), then sin(y) is equal to:

If y = cot⁻¹(√(cos x)) − tan⁻¹(√(cos x)), then sin(y) is equal to:

• A. tan²(x/2)
• B. cot²(x/2)
• C. sin²(x/2)
• D. cos²(x/2)

Answer: A) tan²(x/2)

Explanation: cot⁻¹(k) = π/2 − tan⁻¹(k). So y = π/2 − 2tan⁻¹(√cos x). sin y = sin(π/2 − 2tan⁻¹(√cos x)) = cos(2tan⁻¹(√cos x)). Let tan⁻¹(√cos x) = t. cos 2t = (1−tan²t)/(1+tan²t) = (1−cos x)/(1+cos x) = tan²(x/2).