Question

If $n$ is an integer, prove that
$\tan \{\frac{n\pi }{2}+{(-1)}^n\frac{\pi }{4}\}=1$

Answer 1

case-1: $n=2k$

$\tan (k\pi +(-1{)}^{2k}\frac{\pi }{4})=\tan (k\pi +\frac{\pi }{4})=1$

case-2: $n=2k+1$

$\tan (k\pi +\frac{\pi }{2}+(-1{)}^{2k+1}\frac{\pi }{4})=\tan (k\pi +\frac{\pi }{4})=1$