Solving $\sin 2x + \cos 2x =-1$ $$\sin 2x + \cos 2x =-1$$


How would I go about solving this equation? Some hint on how to start, so I can try to figure it out on my own.
Thank you
$$\sin 2x+\cos 2x=-1$$
$$2\sin x\cos x+\cos^2x-\sin^2x+\sin^2x+\cos^2x=0$$
$$2\sin x\cos x+2\cos^2x=0$$
$$\cos x(\sin x+\cos x)=0$$
$$\cos x=0\Rightarrow x=\frac{\pi}{2}+k\pi,\tan x=-1\Rightarrow x=-\frac{\pi}{4}+k\pi$$
From $\sin 2x + \cos 2x =-1$ we get with $t=2x$:

$1=\sin^2 t+\cos^2 t+2 \sin t \cos t=1+ \sin(2t)$ .

Hence $ \sin(4x)=0$.