Question

Solve the following equations,${\left(\frac{x+i}{x-i}\right)}^n=1.$

Answer 1

${\left(\frac{x+i}{x-i}\right)}^n=1$

$\Rightarrow {\left(\frac{\sqrt{1+x^2}{e}^{i\theta }}{\sqrt{1+x^2}{e}^{-i\theta }}\right)}^n=1$

$\Rightarrow {\left(e^{i2\theta }\right)}^n=1$

$\Rightarrow e^{i2n\theta }=e^{i2k\theta }\left(kϵI\right)$

$2n\theta =2k\pi$

n=any positive integer