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Question: \[\left( \frac{1 + \sin\theta + i\cos\theta}{1 + \sin\theta - i\cos\theta} \right)^{n} =\]...

(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} =

A

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)

B

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)

C

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)

D

cos⁡n(nπ2+nθ)+isin⁡n(nπ2+nθ)\cos n\left( \frac{n\pi}{2} + n\theta \right) + i\sin n\left( \frac{n\pi}{2} + n\theta \right)

Answer

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)

Explanation

Solution

Sol. (1+sin⁡θ+icos⁡θ1+sin⁡θ−icos⁡θ)n=(1+cos⁡α+isin⁡α1+cos⁡α−isin⁡α)n\left( \frac{1 + \sin\theta + i\cos\theta}{1 + \sin\theta - i\cos\theta} \right)^{n} = \left( \frac{1 + \cos\alpha + i\sin\alpha}{1 + \cos\alpha - i\sin\alpha} \right)^{n}

=(2cos⁡2α2+2isin⁡α2cos⁡α22cos⁡2α2−2isin⁡α2⋅cos⁡α2)n=(cos⁡α2+isin⁡α2cos⁡α2−isin⁡α2)n= \left( \frac { 2 \cos ^ { 2 } \frac { \alpha } { 2 } + 2 i \sin \frac { \alpha } { 2 } \cos \frac { \alpha } { 2 } } { 2 \cos ^ { 2 } \frac { \alpha } { 2 } - 2 i \sin \frac { \alpha } { 2 } \cdot \cos \frac { \alpha } { 2 } } \right) ^ { n } = \left( \frac { \cos \frac { \alpha } { 2 } + i \sin \frac { \alpha } { 2 } } { \cos \frac { \alpha } { 2 } - i \sin \frac { \alpha } { 2 } } \right) ^ { n } =[cis(α2)cis(−α2)]n={cis(α2+α2)}n=cis(nα)=cisn(π2−θ)=cis(nπ2−θ)=cos⁡(nπ2−nθ)+isin⁡(nπ2−nθ).\mathbf{=}\left\lbrack \frac{\mathbf{cis}\left( \frac{\mathbf{\alpha}}{\mathbf{2}} \right)}{\mathbf{cis}\left( \mathbf{-}\frac{\mathbf{\alpha}}{\mathbf{2}} \right)} \right\rbrack^{\mathbf{n}}\mathbf{=}\left\{ \mathbf{cis}\left( \frac{\mathbf{\alpha}}{\mathbf{2}}\mathbf{+}\frac{\mathbf{\alpha}}{\mathbf{2}} \right) \right\}^{\mathbf{n}}\mathbf{= cis(n\alpha) = cisn}\left( \frac{\mathbf{\pi}}{\mathbf{2}}\mathbf{- \theta} \right)\mathbf{= cis}\left( \frac{\mathbf{n\pi}}{\mathbf{2}}\mathbf{- \theta} \right)\mathbf{=}\mathbf{\cos}\left( \frac{\mathbf{n\pi}}{\mathbf{2}}\mathbf{-}\mathbf{n\theta} \right)\mathbf{+ i}\mathbf{\sin}\left( \frac{\mathbf{n\pi}}{\mathbf{2}}\mathbf{-}\mathbf{n\theta} \right)\mathbf{.}