Explanation

(c)
$12\cot^{2}{\theta - 31cosec\ \theta + 32 = 0}$
$\Rightarrow \ \ 12\cos^{2}{\theta - 31\sin{\theta + 32\sin^{2}{\theta = 0}}}$
$\Rightarrow \ 20\sin^{2}{\theta - 31\sin{\theta + 12 = 0\ \ \ \ \lbrack\because\cos^{2}{\theta = 1 - \sin^{2}\theta}\rbrack}}$
This is a quadratic equation in $\sin\theta$
$\therefore\ \sin{\theta = \frac{31 \pm \sqrt{31^{2} - 4.20.12}}{2.20}} = \frac{31 \pm 1}{40}\ \Rightarrow \sin{\theta = \frac{4}{5},\frac{3}{4}}$