A) \[\sin 12\theta -i\cos 12\theta \]
B) \[-\sin 12\theta +i\cos 12\theta \]
C) \[-\sin 12\theta -i\cos 12\theta \]
D) \[\sin 12\theta +i\cos 12\theta \]
Correct Answer: B
Solution :
\[\frac{{{(\cos \theta +i\sin \theta )}^{5}}}{{{(\sin \theta +i\cos \theta )}^{7}}}\] \[={{(\cos \theta +i\sin \theta )}^{5}}{{(\sin \theta +i\cos \theta )}^{-7}}\] \[={{(\cos \theta +i\sin \theta )}^{5}}\left[ \frac{1}{-i}{{(-i\sin \theta -{{i}^{2}}\cos \theta )}^{-7}} \right]\] \[=\frac{1}{-i}{{(\cos \theta +i\sin \theta )}^{5}}{{(\cos \theta -i\sin \theta )}^{-7}}\] \[=\frac{1}{-{{i}^{2}}}{{(\cos \theta +i\sin \theta )}^{5}}{{(\cos \theta +i\sin \theta )}^{-7\times -1}}\] \[=i{{(\cos \theta +i\sin \theta )}^{5+7}}\] \[=i{{(\cos \theta +i\sin \theta )}^{12}}\] \[=i(\cos 12\theta +i\sin 12\theta )\] \[=i\cos 12\theta +{{i}^{2}}\sin 12\theta \] \[=-\sin 12\theta +i\cos 12\theta \]
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