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Question

If [1+cosθ+isinθsinθ+i(1+cosθ)]4=cosnθ+isinnθ, then n is ______.

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Solution

[1+cosθ+isinθsinθ+i(1+cosθ)]4=cosnθ+isinnθ
⎢ ⎢ ⎢2cos2θ2+2isinθ2cosθ22sinθ2cosθ2+2icos2θ2⎥ ⎥ ⎥4=cosnθ+isinnθ
⎢ ⎢ ⎢ ⎢cosθ2+isinθ2i(cosθ2isinθ2)⎥ ⎥ ⎥ ⎥4=cosnθ+isinnθ
(cosθ+isinθ)4=cosnθ+isinnθ
(cos4θ+isin4θ)=cosnθ+isinnθ
Therefore, n=4
Ans: 4

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