Mathematics - Trigonometric Equation Question with Solution | TestHub

$\begin{array}{rl}& 12{\cot }^2\theta -31\mathrm{cosec}\theta +32=0\\ & \Rightarrow 12\left({\mathrm{cosec}}^2\theta -1\right)-31\mathrm{cosec}\theta +32=0\\ & \Rightarrow 12{\mathrm{cosec}}^2\theta -31\mathrm{cosec}\theta +20=0\\ & \Rightarrow 12{\mathrm{cosec}}^2\theta -16\mathrm{cosec}\theta -15\mathrm{cosec}\theta +20=0\\ & \Rightarrow (4\mathrm{cosec}\theta -5)(3\mathrm{cosec}\theta -4)=0\\ & \Rightarrow \mathrm{cosec}\theta =\frac{5}{4},\frac{4}{3};\therefore \sin \theta =\frac{4}{5},\frac{3}{4}\end{array}$