Complex Power/Examples/(1 + i \tan (4m+1 over 4n) pi)^n

Example of Complex Power
For $m, n \in \Z$ such that $n \ne 0$:


 * $\paren {1 + i \map \tan {\dfrac {4 m + 1} {4 n} \pi} }^n = \paren {-1}^m \paren {\sec \dfrac {4 m + 1} {4 n} \pi}^n \paren {\dfrac {1 + i} {\sqrt 2} }$

Proof
First setting $x = \dfrac {4 m + 1} {4 n} \pi$, we have:

Now we have:

The result follows.