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  4. cos ((3 π/4)+x)- cos ((3 π/4)-x) is equal to

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Q. $\cos \left(\frac{3 \pi}{4}+x\right)-\cos \left(\frac{3 \pi}{4}-x\right)$ is equal to

Trigonometric Functions

Solution:

$\cos \left(\frac{3 \pi}{4}+x\right)-\cos \left(\frac{3 \pi}{4}-x\right)$
$=-2 \sin \frac{\frac{3 \pi}{4}+x+\frac{3 \pi}{4}-x}{2} \sin \frac{\frac{3 \pi}{4}+x-\frac{3 \pi}{4}+x}{2}$
$=-2 \sin \frac{3 \pi}{4} \sin x$
$=-2 \sin \left(\pi-\frac{\pi}{4}\right) \sin x$
$=-2 \sin \frac{\pi}{4} \sin x=-2 \times \frac{1}{\sqrt{2}} \sin x=-\sqrt{2} \sin x$