Question

Prove that:

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

Answer 1

We have

L.H.S = $2{\text{sin}}^2\frac{3\pi }{4}+2{\text{cos}}^2\frac{\pi }{4}+2{\text{sec}}^2\frac{\pi }{3}$

Now sin $\frac{\pi }{4}$= sin $\left(\frac{\pi -\pi }{4}\right)$ = sin $\frac{\pi }{4}=\frac{1}{\sqrt{2}}$

$\therefore 2{\text{sin}}^2\frac{3\pi }{4}+2{\text{cos}}^2\frac{\pi }{4}+2{\text{sec}}^2\frac{\pi }{3}$

= $2\times {\left(\frac{1}{\sqrt{2}}\right)}^2+2\times {\left(\frac{1}{\sqrt{2}}\right)}^2+2\times (2{)}^2$

= $2\times \frac{1}{2}+2\times \frac{1}{2}+2\times 4$

= 1+ 1+ 8 = 10 = R.H.S