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