Question

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

Answer 1

$\frac{\cos 4\alpha +1}{\cot \alpha -\tan \alpha }$

$=\frac{2{\cos }^2\left(2\alpha \right)}{\frac{\cos \alpha }{\sin \alpha }-\frac{\sin \alpha }{\cos \alpha }}[\because \cos 4\alpha =2{\cos }^2\left(2\alpha \right)-1\&\cos 4\alpha +1=2co{s}^2\left(2\alpha \right)]$

$=\frac{2{\cos }^2\left(2\alpha \right)}{\frac{{\cos }^2\alpha {\sin }^2\alpha }{\sin \alpha \cos \alpha }}$

$=\frac{\left(2\sin \alpha \cos \alpha \right){\cos }^2\left(2\alpha \right)}{\cos 2\alpha }$

$=\sin 2\alpha \cos 2\alpha$

$=\frac{2\sin 2\alpha \cos 2\alpha }{2}=\frac{\sin 4\alpha }{2}$