<s>[INST] 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
 [/INST] Use sum to product for $\sin x + \sin y$ (it works also for $\cos x + \cos y$):
$$\begin{align}
\sin 2x + \cos 2x&=\sin 2x + \sin\left(\frac{\pi}{2}-2x\right)\\
&=2\sin\frac{2x+\pi/2-2x}{2}\cos\frac{2x-\pi/2+2x}{2}\\
&=2\sin\frac{\pi}{4}\cos\left(2x-\frac{\pi}{4}\right)\\
&=\sqrt 2\cos\left(2x-\frac{\pi}{4}\right)
\end{align}$$
The rest should be easy.
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