Question

prove the following
${\cos }^2\left(\frac{\pi }{10}\right)+{\cos }^2\left(\frac{2\pi }{5}\right)+{\cos }^2\left(\frac{3\pi }{5}\right)+{\cos }^2\left(\frac{9\pi }{10}\right)=2$

Answer 1

Solving given question,we get
$co{s}^2\frac{\pi }{10}$ + $co{s}^2[\pi -\frac{2\pi }{5}]$ + $co{s}^2[\pi -\frac{\pi }{10}]$
${\cos }^2\frac{\pi }{10}$+ ${\cos }^2\frac{2\pi }{5}$+ ${\cos }^2\frac{2\pi }{5}$+ ${\cos }^2\frac{\pi }{10}$
=> 2 [ ${\cos }^2\frac{\pi }{10}$+ ${\cos }^2\frac{2\pi }{5}$]
=> 2 [ ${\cos }^2\frac{\pi }{10}$+ ${\cos }^2(\frac{\pi }{2}-\frac{\pi }{10})$]
=> 2 [ ${\cos }^2\frac{\pi }{10}$+ ${\sin }^2\frac{\pi }{10}$]
=> $2$
Hence proved.