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Evaluate $\sin^2\large\frac{\pi}{8}+$$\sin^2\large\frac{3\pi}{8}$$+\sin^2\large\frac{5\pi}{8}$$+\sin^2\large\frac{7\pi}{8}$

$(a)\;1\qquad(b)\;2\qquad(c)\;3\qquad(d)\;4$

1 Answer

$\sin\large\frac{7\pi}{8}$$=\sin(\pi-\large\frac{\pi}{8})$
$\quad\quad\;\;=\sin\large\frac{\pi}{8}$
$\sin\large\frac{5\pi}{8}$$=\sin(\pi-\large\frac{3\pi}{8})$
$\quad\quad\;\;=\sin\large\frac{3\pi}{8}$
$\sin\large\frac{3\pi}{8}$$=\sin(\large\frac{\pi}{2}-\large\frac{3\pi}{8})$
$\quad\quad\;\;=\cos\large\frac{\pi}{8}$
Now we can get
$\sin^2\large\frac{\pi}{8}$$+\cos^2\large\frac{\pi}{8}$$+\sin^2\large\frac{3\pi}{8}+$$\sin^2\large\frac{\pi}{8}$
$\Rightarrow \sin^2\large\frac{\pi}{8}$$+\sin^2\large\frac{3\pi}{8}$$+\sin^2\large\frac{3\pi}{8}+$$\sin^2\large\frac{\pi}{8}$
$\Rightarrow 2[\sin^2\large\frac{\pi}{8}+$$\sin^2\large\frac{3\pi}{8}]$
$\Rightarrow 2[\sin^2\large\frac{\pi}{8}+$$\cos^2\large\frac{\pi}{8}]$
$\Rightarrow 2[1]$
$\Rightarrow 2$
Hence (b) is th correct answer.
answered Oct 10, 2013 by sreemathi.v
 

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