<s>[INST] Prove $\cos((3\pi/4)+x)-\cos((3\pi/4)-x)=(-\sqrt{2})\sin(x)$ Please help me to solve the below trigonometric function as i am trying it from the last hour.
$$\cos((3\pi/4)+x)-\cos((3\pi/4)-x)=(-\sqrt{2})\sin(x)$$
 [/INST] Apply, $\cos(A+B)-\cos(A-B)=-2\sin A \sin B$
So, $\cos(3\frac{\pi}{4}+x)-\cos(3\frac{\pi}{4}-x)=-2\sin x\sin(3\frac{\pi}{4}) $
$=-2\sin x\sin(\pi-\frac{\pi}{4})=-2\sin x\sin(\frac{\pi}{4})$
 as $\sin(\pi-y)=\sin y$
So, $\cos(3\frac{\pi}{4}+x)-\cos(3\frac{\pi}{4}-x)=-\sqrt2\sin x$
or apply $\cos 2C- \cos 2D=-2sin(C+D)sin(C-D)$
Put $2C=3\frac{\pi}{4}+x, 2D=3\frac{\pi}{4}-x=>C+D=3\frac{\pi}{4}$ and $C-D=x$
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