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  4. If L= sin 2((π/16))- sin 2((π/8)) and M= cos 2((π/16))- sin 2((π/8)), then :

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Q. If $L=\sin ^{2}\left(\frac{\pi}{16}\right)-\sin ^{2}\left(\frac{\pi}{8}\right)$ and $M=\cos ^{2}\left(\frac{\pi}{16}\right)-\sin ^{2}\left(\frac{\pi}{8}\right),$ then :

JEE MainJEE Main 2020Trigonometric Functions

Solution:

$L=\sin ^{2}\left(\frac{\pi}{16}\right)-\sin ^{2}\left(\frac{\pi}{8}\right)$
$\left(\because \sin ^{2} \theta=\frac{1-\cos 2 \theta}{2}\right)$
$\Rightarrow L=\left(\frac{1-\cos (\pi / 8)}{2}\right)-\left(\frac{1-\cos (\pi / 4)}{2}\right)$
$L=\frac{1}{2}\left[\cos \left(\frac{\pi}{4}\right)-\cos \left(\frac{\pi}{8}\right)\right]$
$L=\frac{1}{2 \sqrt{2}}-\frac{1}{2} \cos \left(\frac{\pi}{8}\right)$
$M=\cos ^{2}\left(\frac{\pi}{16}\right)-\sin ^{2}\left(\frac{\pi}{8}\right)$
$M=\frac{1+\cos (\pi / 8)}{2}-\frac{1-\cos (\pi / 4)}{2}$
$M=\frac{1}{2} \cos \left(\frac{\pi}{8}\right)+\frac{1}{2 \sqrt{2}}$