Question

If $12{\cot }^2\theta -31cosec\theta +32=0$ then value of $\sin \theta$ is :

• A. $\frac{3}{5}$ or $1$
• B. $\frac{2}{3}$ or $\frac{-2}{3}$
• C. $\frac{4}{5}$ or $\frac{3}{4}$
• D. $\pm \frac{1}{2}$

Answer 1

Answer: (C)

$\frac{4}{5}$ or $\frac{3}{4}$

Let $x=cosec\theta$
$12{\cot }^2\theta -31cosec\theta +32=0$
$\Rightarrow 12(x^2-1)-31x+32=0$
$\Rightarrow 12x^2-31x+20=0$
$\Rightarrow (4x-5)(3x-4)=0$
$x=\frac{5}{4}$ or $\frac{4}{3}$
$\therefore \sin \theta =\frac{4}{5}$ or $\frac{3}{4}$
Hence, option $\left(3\right)$ is correct.