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Question Number 176159 by peter frank last updated on 14/Sep/22

Answered by adhigenz last updated on 14/Sep/22

$$\mathrm{Let}x=\cos \alpha +i.\sin \alpha =e^{i.\alpha }\mathrm{and}y=\cos \beta +i.\sin \beta =e^{i.\beta }$$$$\left(\mathrm{a}\right).x^m{y}^n+\frac{1}{x^m{y}^n}=e^{i.m\alpha }.e^{i.n\beta }+e^{i.m(-\alpha )}.e^{i.n(-\beta )}$$$$=e^{i(m\alpha +n\beta )}+e^{i(-m\alpha -n\beta )}$$$$=\cos (m\alpha +n\beta )+i.\sin (m\alpha +n\beta )+\cos (-m\alpha -n\beta )+i.\sin (-m\alpha -n\beta )$$$$=2\cos (m\alpha +n\beta )$$

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