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Prove that cosec^4θ (1 − cos^4θ) = 1 + 2cot^2θ.

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Prove that cosec4θ (1 − cos4θ) = 1 + 2cot2θ.

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cosec4θ (1 − cos4θ) − 2cot2θ = cosec2 θ(1 - cos2 θ)(1 + cosθ)/sin2θ

= cosec2 θ(1 + cos2 θ) − 2cot2 θ = cosec2 θ + cot2 θ − 2cot2 θ = 1 + 2 cot2 θ − 2cot2 θ = 1

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+1 vote
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cosec^4θ + cosec^2θ = cot^4θ + cot^2θ
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