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  4. If cos ((π/4)-x) cos 2 x+ sin x sin 2 x sec x= cos x sin 2 x sec x+ then a possible value of sec x is

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

TS EAMCET 2020

Solution:

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