Question

If m, n are integers and $x=\cos \alpha +i\sin \alpha$, $y=\cos \beta +\sin \beta$, then find $x^m{y}^n+\frac{1}{x^m{y}^n}$.

Answer 1

$x=\cos \alpha +i\sin \alpha =e^{i\alpha }$

$y=\cos \beta +i\sin \beta =e^{i\beta }$

$x^m{y}^n+\frac{1}{x^m{y}^n}=e^{i(m\alpha +n\beta )}+\frac{1}{e^{i(m\alpha +n\beta )}}$

$\Rightarrow e^{i(m\alpha +n\beta )}+e^{-i(m\alpha +n\beta )}$

$=\cos (m\alpha +n\beta )+i\sin (m\alpha +n\beta )+\cos (m\alpha +n\beta )-i\sin (m\alpha +n\beta )$

$=2\cos (m\alpha +n\beta )$