If and , then is equal…

MATHEMATICS · Single correct · CBSE · Mathematics · Complex Numbers and Quadratic Equations · Complex Numbers and Quadratic Equations

Problem

If θ∈R\theta \in \mathbb{R} and i=−1i = \sqrt{-1} , then (1+sin⁡θ+icos⁡θ1+sin⁡θ−icos⁡θ)n\left(\frac{1 + \sin \theta + i \cos \theta}{1 + \sin \theta - i \cos \theta}\right)^n is equal to (1) cos⁡(nπ2−nθ)+isin⁡(nπ2−nθ)\cos\left(\frac{n\pi}{2} - n\theta\right) + i \sin\left(\frac{n\pi}{2} - n\theta\right) (2) cos⁡(nπ2+nθ)+isin⁡(nπ2+nθ)\cos\left(\frac{n\pi}{2} + n\theta\right) + i \sin\left(\frac{n\pi}{2} + n\theta\right) (3) sin⁡(nπ2−nθ)+icos⁡(nπ2−nθ)\sin\left(\frac{n\pi}{2} - n\theta\right) + i \cos\left(\frac{n\pi}{2} - n\theta\right) (4) cos⁡(n(π2+2θ))+isin⁡(n(π2+2θ))\cos\left(n\left(\frac{\pi}{2} + 2\theta\right)\right) + i \sin\left(n\left(\frac{\pi}{2} + 2\theta\right)\right)

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