Complex Numbers And Quadratic Equations question 147
Question: 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
Options:
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 ) $
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Answer:
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 ) $
BETA
