If $x+\large\frac{1}{x}$$=2\cos\;\theta$ and $y+\large\frac{1}{y}=$$2 \cos \;\phi$ show that $\large\frac{x^{m}}{y^{n}}+ \frac{y^{n}}{x^{m}}$$= 2 \cos \left ( m\theta -n\phi \right )$

Question

where m, n ∈ N.

This is the first part of the multi-part question Q8.