A) \[\cos \left( \frac{n\pi }{2}-n\theta \right)+i\,\sin \left( \frac{n\pi }{2}-n\theta \right)\]
B) \[\cos \left( \frac{n\pi }{2}+n\theta \right)+i\,\sin \left( \frac{n\pi }{2}+n\theta \right)\]
C) \[\sin \left( \frac{n\pi }{2}-n\theta \right)+i\,\cos \left( \frac{n\pi }{2}-n\theta \right)\]
D) \[\cos \,n\left( \frac{\pi }{2}+2\theta \right)+i\,\sin \,n\left( \frac{\pi }{2}+2\theta \right)\]
Correct Answer: A
Solution :
\[{{\left( \frac{1+\sin \theta +i\cos \theta }{1+\sin \theta -i\cos \theta } \right)}^{n}}={{\left( \frac{1+\cos \alpha +i\sin \alpha }{1+\cos \alpha -i\sin \alpha } \right)}^{n}}\]\[(\]where \[\alpha =\frac{\pi }{2}-\theta )\] \[={{\left( \frac{2{{\cos }^{2}}\frac{\alpha }{2}+2i\sin \frac{\alpha }{2}\cos \frac{\alpha }{2}}{2{{\cos }^{2}}\frac{\alpha }{2}-2i\sin \frac{\alpha }{2}\cos \frac{\alpha }{2}} \right)}^{n}}\]\[={{\left( \frac{\cos \frac{\alpha }{2}+i\sin \frac{\alpha }{2}}{\cos \frac{\alpha }{2}-i\sin \frac{\alpha }{2}} \right)}^{n}}\] \[={{\left( \frac{cis\,\left( \frac{\alpha }{2} \right)}{cis\,\left( -\frac{\alpha }{2} \right)} \right)}^{n}}\]\[={{\left\{ cis\,\left( \frac{\alpha }{2}+\frac{\alpha }{2} \right) \right\}}^{n}}=cis(n\alpha )\] = \[cis\,n\,\left( \frac{\pi }{2}-\theta \right)=cis\,\left( \frac{n\pi }{2}-n\theta \right)\] \[=\,\cos \left( \frac{n\pi }{2}-n\theta \right)+i\,\sin \,\left( \frac{n\pi }{2}-n\theta \right)\].
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