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Question
Prove the following identity:
1 + 3cosec2θ cot2θ + cot6θ = cosec6θ
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Solution
L.H.S. = 1 + 3cosec2θ cot2θ + cot6θ
= 1 + 3 cosec2θ cot2θ + (cot2θ)3
= 1 + 3 cosec2θ (cosec2θ – 1) + (cosec2θ – 1)3 ...`[(1 + cot^2theta = cosec^2theta),(cot^2theta = cosec^2 theta - 1)]`
= 1 + 3 cosec4θ – 3 cosec2θ + cosec6θ – 3 cosec4θ + 3cosec2θ – 1 ...[(a - b)3 = a3 - 3a2b + 3ab2 - b3]
= `cancel(1) + cancel(3 cosec^4θ) - cancel(3 cosec^2θ) + cosec^6θ - cancel(3 cosec^4 θ) + cancel(3cosec^2 θ) cancel(- 1)`
= cosec6θ
= R.H.S.
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