<s>[INST] $-\sin x -\cos x = -\sqrt{2}\sin (x+{\pi\over4})$ 
$-\sin x -\cos x = -\sqrt{2}\sin (x+{\pi\over4})$

How does the cosine disappear and how did sin x turn into $\sin (x+{\pi\over4})$?
 [/INST] $$\sin x\cos a + \cos x \ sin a = \sin(x+a)$$
$$-\sin x -\cos x = - (\sin x +\cos x ) = -(\sin x\cos a /cos a + \cos x \ sin a / \ sin a)$$
choosing $\cos a = \ sin a$ , which means $a = \pi/4$, we have
$$-(\sin x\cos a /cos a + \cos x \ sin a / \ sin a) = -\sin(x+\pi/4)\times 2/\sqrt{2}  $$
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