Step 1:
$x=\cos\alpha+i\sin\alpha$
$y=\cos\beta+i\sin\beta$
$\Rightarrow x=e^{\large i\alpha}$
$y=e^{\large i\beta}$
$x^my^n+\large\frac{1}{x^my^n}$$=(e^{i\alpha})^m(e^{i\beta})^n+\large\frac{1}{(e^{i\alpha})^m(e^{i\beta})}$
$\qquad\qquad\;\;\;=e^{im\alpha}e^{im\beta}+\large\frac{1}{e^{\Large im\alpha}e^{\Large im\beta}}$
$\qquad\qquad\;\;\;=e^{\large i(m\alpha+n\beta)}+e^{\large-i(m\alpha+n\beta)}$
$\qquad\qquad\;\;\;=\cos(m\alpha+n\beta)+i\sin(m\alpha+n\beta)+(\cos(m\alpha+n\beta)-i\sin(m\alpha+\beta))$
$\qquad\qquad\;\;\;=2\cos(m\alpha+n\beta)$