For 0 < θ < π/2, consider the following: I. (tan^4θ + tan^6θ)(cot^4θ + cot^6θ) = sec^2θ cosec^2θ II. (tanθ + sinθ)/(tanθ - sinθ) = cot^2θ (secθ + 1)^2 Which of the above is/are identities?

CDS 2026 PYQ with Answer — For 0 < θ < π/2, consider the following: I. (tan^4θ + tan^6θ)(cot^4θ…

2026 · Q42

For 0 < θ < π/2, consider the following:

I(tan^4θ + tan^6θ)(cot^4θ + cot^6θ) = sec^2θ cosec^2θ
II(tanθ + sinθ)/(tanθ - sinθ) = cot^2θ (secθ + 1)^2

Which of the above is/are identities?

Elementary MathematicsTrigonometry

AI only
BII only
CBoth I and II
DNeither I nor II

Explanation

For I: tan⁴θ+tan^6θ=tan⁴θ·sec²θ and cot⁴θ+cot^6θ=cot⁴θ·cosec²θ, whose product is sec²θ·cosec²θ — an identity. For II: (tanθ+sinθ)/(tanθ−sinθ)=(1+cosθ)/(1−cosθ), and cot²θ(secθ+1)²=(1+cosθ)²/sin²θ, which equals the same — also an identity. Both are identities, option C.