The value of cos[tan⁻¹{sin(cot⁻¹x)}] is:

The value of cos[tan⁻¹{sin(cot⁻¹x)}] is:

• A. √((x²+1)/(x²+2))
• B. √((x²+2)/(x²+1))
• C. x/√(x²+2)
• D. 1/√(x²+2)

Answer: A) √((x²+1)/(x²+2))

Explanation: Let cot⁻¹x = θ, so cot θ = x → sin θ = 1/√(1+x²). Then tan⁻¹(sin θ) = tan⁻¹(1/√(1+x²)) = φ, where tan φ = 1/√(1+x²). We need cos φ = 1/√(1+tan²φ) = 1/√(1+1/(1+x²)) = √((1+x²)/(2+x²)). Yes, matches option 0.