A) \[1\]
B) \[\frac{1}{2}\]
C) \[\frac{1}{2}\cos \left( \frac{\pi }{10} \right)\]
D) \[\frac{1}{2}\cos \left( \frac{\pi }{5} \right)\]
E) \[\frac{1}{2}\sec \left( \frac{\pi }{10} \right)\]
Correct Answer: B
Solution :
\[{{\left[ 1+\cos \frac{\pi }{5}+i\sin \frac{\pi }{5} \right]}^{-1}}\]\[=\frac{1}{2{{\cos }^{2}}\frac{\pi }{10}+2i\sin \frac{\pi }{10}.\cos \frac{\pi }{10}}\] \[=\frac{1}{2\cos \frac{\pi }{10}}{{\left[ \cos \frac{\pi }{10}+i\sin \frac{\pi }{10} \right]}^{-1}}\] \[\therefore \]Real part of\[{{\left[ 1+\cos \frac{\pi }{5}+isin\frac{\pi }{10} \right]}^{-1}}\]is\[\frac{1}{2}\].
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