Question

Solve the following equation:
$\frac{1+\sin x+\cos x+\sin 2x+\cos 2x}{\tan 2x}=0$

Answer 1

$1+\sin x+\cos x+2\sin x\cos x+{2\cos }^{2}x-1=0$

$2\cos x(\sin x+\cos x)+1(\sin x+\cos x)=0$

$(\sin x+\cos x)(2\cos x+1)=0$

$\tan x=-1$ and $\cos x=-\frac{1}{2}$

So, $x=n\pi -\frac{\pi }{4}$ and $x=2n\pi \pm \frac{2\pi }{3}.$