Question

X=cos alpa+isin alpha, y=cos beta+isin beta then xpower m*ypower n

Answer 1

Demovire's theorem:

For any natural number n,

$\boxed{(cos\,\theta+i\,sin\theta)^n=cos\,n\theta+i\,sin\,\theta}$

Euler's formula:

$\boxed{cos\,\theta+i\,sin\theta=e^{i\,\theta}}$

Given:

$x=cos\,\alpha+i\,sin\,\alpha$

$y=cos\,\beta+i\,sin\,\beta$

Now,

$x^m\,y^n$

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

Using demovire's theorem

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

Using euler's formula

$=e^{i\,m\alpha}\,e^{i\,n\beta}$

$=e^{i(m\,\alpha+n\,\beta)}$

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

$\therefore\boxed{\bf\,x^m\,y^n=cos(m\,\alpha+n\,\beta)+i\,sin(m\,\alpha+n\,\beta)}$