A) \[\cos 49\theta -i\sin 49\theta \]
B) \[\cos 23\theta +i\sin 23\theta \]
C) \[\cos 49\theta +i\sin 49\theta \]
D) \[\cos 21\theta +i\sin 21\theta \]
Correct Answer: A
Solution :
\[\frac{{{(\cos 2\theta -i\sin 2\theta )}^{4}}{{(\cos 4\theta +i\sin 4\theta )}^{-5}}}{{{(\cos 3\theta +i\sin 3\theta )}^{-2}}{{(\cos 3\theta -i\sin 3\theta )}^{-9}}}\] \[=\frac{{{(\cos \theta -i\sin \theta )}^{8}}{{(\cos \theta +i\sin \theta )}^{-20}}}{{{(\cos \theta +i\sin \theta )}^{-6}}{{(\cos \theta -i\sin \theta )}^{-27}}}\] \[=\frac{{{(\cos \theta +i\sin \theta )}^{-8}}{{(\cos \theta +i\sin \theta )}^{-20}}}{{{(\cos \theta +i\sin \theta )}^{-6}}{{(\cos \theta +i\sin \theta )}^{27}}}\] \[={{(cos\theta +sin\theta )}^{-27-8-20+6}}\] \[={{(\cos \theta +i\sin \theta )}^{-49}}=\cos 49-i\sin 49\]
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