Question

Simplify the following expression:
$\frac{1+\sin 4\alpha -\cos 4\alpha }{1+\cos 4\alpha +\sin 4\alpha }$

Answer 1

$1+\sin 4\alpha -\cos 4\alpha =(\sin 2\alpha +\cos 2\alpha {)}^2-(\cos 2\alpha -\sin 2\alpha \left)\right(\cos 2\alpha +\sin 2\alpha )=2\sin 2\alpha (\sin 2\alpha +\cos 2\alpha )$

$1+\cos 4\alpha +\sin 4\alpha =2{\cos }^{2}2\alpha +2\sin 2\alpha \cos 2\alpha =2\cos 2\alpha (\cos 2\alpha +\sin 2\alpha )$

$\frac{1+\sin 4\alpha -\cos 4\alpha }{1+\cos 4\alpha +\sin 4\alpha }=\tan 2\alpha$