Question

Prove that: $2{\sin }^2\frac{3\pi }{4}+2{\cos }^2\frac{\pi }{4}+2{\sec }^2\frac{\pi }{3}=10$

Answer 1

$L.H.S=2{\sin }^2\frac{3\pi }{4}+2{\cos }^2\frac{\pi }{4}+2{\sec }^2\frac{\pi }{3}$
$=2{\left[\sin (\pi -\frac{\pi }{4})\right]}^2+2{\cos }^2\frac{\pi }{4}+2{\sec }^2\frac{\pi }{3}$
$=2{\sin }^2\frac{\pi }{4}+2{\cos }^2\frac{\pi }{4}+2{\sec }^2\frac{\pi }{3}$
$=2{\left(\frac{1}{\sqrt{2}}\right)}^2+2{\left(\frac{1}{\sqrt{2}}\right)}^2+2(2{)}^2$
$=2[\frac{1}{2}+\frac{1}{2}+4]$
$=2\left(5\right)=10=R.H.S.$

So, $L.H.S.=R.H.S.$
$2{\sin }^2\frac{3\pi }{4}+2{\cos }^2\frac{\pi }{4}+2{\sec }^2\frac{\pi }{3}=10$
Hence proved