Question

$\frac{x^m}{y^n}+\frac{y^n}{x^m}=2cos(m\theta -n\varphi )$

Answer 1

$2\cos \theta =x+\frac{1}{x}$

$\Rightarrow x^2-2x\cos \theta +1=0$

$x=\frac{2\cos \theta \pm \sqrt{4{\cos }^2\theta -4}}{2}$

$=\cos \theta \pm i\sin \theta$

Similarly $y=\cos \varphi \pm i\sin \varphi$

$\frac{x^m}{y^n}+\frac{y^m}{x^n}=x^m{y}^{-n}+y^m{x}^{-n}$

$=(\cos m\theta \pm i\sin m\theta )(\cos n\varphi \mp i\sin n\varphi )+(\cos m\varphi \pm i\sin m\varphi )(\cos n\theta \mp i\sin n\theta )$

$\frac{x^m}{y^n}+\frac{y^m}{x^n}=2Re(\cos m\theta \pm i\sin m\theta )(\cos n\varphi \mp i\sin n\varphi )$

$=2\left(\cos m\theta \cos n\varphi \right(\pm \times \mp \times i^2\left)\sin m\theta \sin n\varphi \right)$

$=2(\cos m\theta \cos n\varphi +\sin m\theta \sin n\varphi )$

$=2\cos (m\theta -n\varphi )$