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Question

Prove that $$cosec^6\theta \, = \, cot^6\theta \, + \, 3cot^2\theta \, cosec^2\theta \, + \, 1$$


Solution

Identity : $$cosec ^{2}\theta - \cot ^{2}\theta =1$$

$$LHS=cosec^6\theta$$ 

          $$=\left (1+\cot^{2}\theta \right) ^{3}$$

          $$=1+\cot ^{6}\theta +3.1.\cot ^{2}\theta (1+\cot^{2}\theta) $$

          $$=\cot ^{6}\theta +3\cot ^{2}\theta cosec ^{2}\theta +1$$

          $$=RHS$$ 

Hence proved 

Mathematics

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