A) \[\sin 8\theta -i\cos 8\theta \]
B) \[\cos 8\theta -i\sin 8\theta \]
C) \[\sin 8\theta +i\cos 8\theta \]
D) \[\cos 8\theta +i\sin 8\theta \]
Correct Answer: D
Solution :
\[{{\left( \frac{\cos \,\theta +i\sin \theta }{\sin \,\theta +i\cos \,\,\theta } \right)}^{4}}=\frac{{{(\cos \,\,\theta +i\sin \,\theta )}^{4}}}{{{i}^{4}}\,{{(\cos \,\,\theta -i\sin \theta )}^{4}}}\]\[=\frac{\cos \,\,4\theta +i\sin \,\,4\theta }{\cos \,\,4\theta -i\sin \,\,4\theta }\] \[=\frac{(\cos 4\theta +i\sin 4\theta )(\cos 4\theta +i\sin 4\theta )}{(\cos 4\theta -i\sin 4\theta )(\cos 4\theta +i\sin 4\theta )}\] \[\frac{{{(\cos 4\theta +i\sin 4\theta )}^{2}}}{{{\cos }^{2}}4\theta +{{\sin }^{2}}4\theta }=\cos 8\theta +i\sin 8\theta \]
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