Question

If $\cos \left(\frac{\pi }{4}-x\right)\cos 2x+\sin x\sin 2x\sec x=\cos x\sin 2x\sec x+$ then a possible value of $\sec x$ is

• A. $\frac{1}{2\sqrt{2}}$
• B. $3\sqrt{2}$
• C. $\frac{1}{\sqrt{2}}$
• D. $\sqrt{2}$

Answer 1

Answer: (D)

Given, $\cos \left(\frac{\pi }{4}-x\right)\cos 2x+\sin x\sin 2x\sec x$
$=\cos x\sin 2x\sec x+\cos \left(\frac{\pi }{4}+x\right)\cos 2x$
$\cos 2x\left[\cos \left(\frac{\pi }{4}-x\right)+\cos \left(\frac{\pi }{4}+x\right)\right]$
$\sec x\sin 2x(\cos x-\sin x)$
$\cos 2x\left(2\sin \frac{\pi }{4}\sin x\right)$
$=\sec x\sin 2x(\cos x-\sin x)$
$\frac{2}{\sqrt{2}}\left({\cos }^{2}x-{\sin }^{2}x\right)\sin x$
$=\sec x(2\sin x\cos x)$
$\cos x+\sin x=\sqrt{2}$
$(\cos x-\sin x)$
$\because \sin x=\cos x=\frac{1}{\sqrt{2}}$
$\sec x=\sqrt{2}$