Question

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

• A. $\sqrt{2}\sin x$
• B. $-2\sin x$
• C. $-\sqrt{2}\sin x$
• D. None of these

Answer 1

Answer: (C)

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