QPaperGen

We know the standard values:
\[
\sin\frac{\pi}{6} = \frac{1}{2}, \quad \cos\frac{\pi}{3} = \frac{1}{2}.
\]

Also,
\[
\text{cosec}\left(\frac{7\pi}{6}\right) = \text{cosec}\left(\pi + \frac{\pi}{6}\right) = \text{cosec}\left(\frac{\pi}{6}\right) = -2.
\]

Therefore,
\[
\text{LHS} = 2\sin^2\frac{\pi}{6} + \text{cosec}^2\frac{7\pi}{6}\,\cos^2\frac{\pi}{3}
\]

\[
= 2\left(\frac{1}{2}\right)^2 + \left((-2)^2\right)\left(\frac{1}{2}\right)^2
\]

\[
= 2\cdot \frac{1}{4} + 4\cdot \frac{1}{4}
\]

\[
= \frac{1}{2} + 1 = \frac{3}{2}.
\]

\[
\therefore \text{LHS} = \text{RHS}.
\]

Hence proved.