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3.Trigonometrical Ratios, Functions and Identities
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Value of ${\sin ^2}\frac{\pi }{8} + {\sin ^2}\frac{{3\pi }}{8} + {\sin ^2}\frac{{5\pi }}{8} + {\sin ^2}\frac{{7\pi }}{8}$ is
A
$1$
B
$2$
C
$1\frac{1}{8}$
D
$2\frac{1}{8}$
Solution
$\sin \frac{7 \pi}{8}=\sin \left(\pi-\frac{\pi}{8}\right)=\sin \frac{\pi}{8}$
$\sin \frac{5 \pi}{8}=\sin \left(\pi-\frac{3 \pi}{8}\right)=\sin \frac{3 \pi}{8}$
$\therefore $ $\text { The given value } =2\left[\sin ^{2} \frac{\pi}{8}+\sin ^{2} \frac{3 \pi}{8}\right] $
$=2\left[\sin ^{2} \frac{\pi}{8}+\cos ^{2} \frac{\pi}{8}\right] $
$\left[\because \sin \frac{3 \pi}{8}\right.\left.=\sin \left(\frac{\pi}{2}-\frac{\pi}{8}\right)=\cos \frac{\pi}{8}\right] $
$=2(1)=2$
Standard 11
Mathematics
