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Question
Prove the following:
2 sec2θ – sec4θ – 2cosec2θ + cosec4θ = cot4θ – tan4θ
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Solution
L.H.S. = 2 sec2θ – sec4θ – 2cosec2θ + cosec4θ
= 2 sec2θ – (sec2θ)2 – 2cosec2θ + (cosec2θ)2
= 2(1 + tan2θ) – (1 + tan2θ)2 – 2(1 + cot2θ) + (1 + cot2θ)2
= 2 + 2tan2θ – (1 + 2tan2θ + tan4θ) – 2 – 2cot2θ + 1 + 2cot2θ + cot4θ
= 2 + 2tan2θ – 1 – 2tan2θ – tan4θ – 2 – 2cot2θ + 1 + 2cot2θ + cot4θ
= cot4θ – tan4θ
= R.H.S.
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