question_answer

If \[x=\cos \theta +i\sin \theta \] and \[y=\cos \varphi +i\sin \varphi \],  then \[{{x}^{m}}{{y}^{n}}+{{x}^{-m}}{{y}^{-n}}\] is equal to

A) \[\cos (m\theta +n\varphi )\]

B) \[\cos (m\theta +n\varphi )\]

C) \[2\cos (m\theta +n\varphi )\]

D) \[2\cos (m\theta -n\varphi )\]

Correct Answer: C

Solution :

\[x=\cos \theta +i\sin \theta ={{e}^{i\theta }},y=\cos \varphi +i\sin \varphi ={{e}^{i\varphi }}\] \[\therefore \,\,\,\,\,{{x}^{m}}{{y}^{n}}+{{x}^{-m}}{{y}^{-n}}={{e}^{im\theta }}{{e}^{in\varphi }}+{{e}^{-im\theta }}{{e}^{-in\varphi }}\] \[={{e}^{i(m\theta +n\varphi )}}+{{e}^{-i\,\,(m\theta \,+n\varphi )}}\]\[=\cos (m\theta +n\varphi )+i\sin (m\theta +n\varphi )\] \[=2\cos (m\theta +n\varphi )\]