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}}$$ = 2i \;sin\left ( m\theta -n\phi \right )$

Question

where m, n ∈ N.

This is the second part of the Q8.