Explanation

(c)
$12\cot^{2}{\theta - 31\ cosec\ \theta + 32 = 0}$
$\Rightarrow \ \ 12\cos^{2}{\theta - 31\sin{\theta + 32\sin^{2}{\theta = 0}}}$
$\Rightarrow 20\sin^{2}{\theta - 31\sin{\theta + 12 = 0\ \ \ \ \ \left\lbrack \because\cos^{2}{\vartheta = 1 - \sin^{2}\theta} \right\rbrack}}$
$\therefore\ \sin{3\theta = \frac{31 \pm \sqrt{31^{2} - 4.20.12}}{2.20}}$
$= \frac{31 \pm \sqrt{961 - 960}}{40} = \frac{31 \pm 1}{40}$
$\Rightarrow \ \sin{\theta = \frac{4}{5},\frac{3}{4}}$