Question

If $12{\cot }^2\theta -31\text{cosec}\theta +32=0$, then the 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)

$12co{t}^2\theta -31cosec\theta +32=0$
$\Rightarrow 12\left(cose{c}^2\theta -1\right)-31cosec\theta +32=0$
$\Rightarrow 12cose{c}^2\theta -31cosec\theta +20=0$
$\Rightarrow 12cose{c}^2\theta -16cosec\theta -15cosec\theta +20=0$
$\Rightarrow \left(4cosec\theta -5\right)\left(3cosec\theta -4\right)=0$
$\Rightarrow cosec\theta =\frac{5}{4},\frac{4}{3}\therefore sin\theta =\frac{4}{5},\frac{3}{4}.$