Question

Prove that:
$2si{n}^2\frac{\pi }{6}+cose{c}^2\frac{7\pi }{6}co{s}^2\frac{\pi }{3}=\frac{3}{2}$

Answer 1

LHS$=2si{n}^2\frac{\pi }{6}+cose{c}^2\frac{7\pi }{6}co{s}^2\frac{\pi }{3}$

$=2si{n}^2\frac{\pi }{6}+cose{c}^2(\pi +\frac{\pi }{6})co{s}^2\frac{\pi }{3}$

$=2si{n}^2\frac{\pi }{6}+cose{c}^2(-\frac{\pi }{6})co{s}^2\frac{\pi }{3}$ ....As $\theta$ lies in the 3rd quadrant, $\theta$ is negative.

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

$=2\times \frac{1}{4}+1$ $=\frac{3}{2}$

$=$ RHS

Hence proved.