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  4. sin 2 (π /8)+ sin 2 (3π /8)+ sin 2(5π /8)+ sin 2(7π /8) is equal to

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Q. $ {{\sin }^{2}}\,\frac{\pi }{8}+{{\sin }^{2}}\,\frac{3\pi }{8}+{{\sin }^{2}}\frac{5\pi }{8}+{{\sin }^{2}}\frac{7\pi }{8} $ is equal to

J & K CETJ & K CET 2010

Solution:

$ {{\sin }^{2}}\frac{\pi }{8}+{{\sin }^{2}}\frac{3\pi }{8}+{{\sin }^{2}}\frac{5\pi }{8}+{{\sin }^{2}}\frac{7\pi }{8} $
$ {{\sin }^{2}}\frac{\pi }{8}+{{\sin }^{2}}\frac{3\pi }{8}+{{\sin }^{2}}\left( \pi -\frac{3\pi }{8} \right)+{{\sin }^{2}}\left( \pi -\frac{\pi }{8} \right) $
$ {{\sin }^{2}}\frac{\pi }{8}+{{\sin }^{2}}\frac{3\pi }{8}+{{\sin }^{2}}\frac{3\pi }{8}+{{\sin }^{2}}\frac{\pi }{8} $
$ =2\left[ {{\sin }^{2}}\,\frac{\pi }{8}+{{\sin }^{2}}\frac{3\pi }{8} \right] $
$ =2\left[ {{\sin }^{2}}\frac{\pi }{8}+{{\sin }^{2}}\left( \frac{\pi }{2}-\frac{\pi }{8} \right) \right] $
$ =2\left[ {{\sin }^{2}}\frac{\pi }{8}+{{\cos }^{2}}\frac{\pi }{8} \right] $
$ =2.1=2 $