A) \[\cos 9\theta -i\sin 9\theta \]
B) \[\cos \theta -i\sin \theta \]
C) \[\sin 9\theta +i\cos 9\theta \]
D) \[\sin 9\theta -i\cos 9\theta \]
Correct Answer: D
Solution :
\[\frac{{{(\cos \theta +i\sin \theta )}^{4}}}{{{(\sin \theta +i\cos \theta )}^{5}}}=\frac{{{(\cos \theta +i\sin \theta )}^{4}}}{{{\left[ i\left( \cos \theta +\frac{1}{i}\sin \theta \right) \right]}^{5}}}\] \[=\frac{{{(\cos \theta +i\sin \theta )}^{4}}}{{{i}^{5}}{{(\cos \theta -i\sin \theta )}^{5}}}\] \[=\frac{1}{i}{{(\cos \theta +i\sin \theta )}^{4}}.{{(\cos \theta -i\sin \theta )}^{-5}}\] \[=\frac{1}{i}{{(\cos \theta +i\sin \theta )}^{4}}.{{(\cos \theta +i\sin \theta )}^{5}}\] \[=\frac{1}{i}{{(\cos \theta +i\sin \theta )}^{9}}\] \[=\frac{1}{i}(\cos 9\theta +i\sin 9\theta )\] \[=\frac{1}{{{i}^{2}}}(i\cos 9\theta +{{i}^{2}}\sin 9\theta )\] \[=(-i\cos 9\theta -\sin 9\theta )\] \[=\sin 9\theta -i\cos 9\theta \]
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