$12{\cot ^2}\theta  – 31\cos ec\theta  + 32 = 0$

$12({\rm{cos}}{\rm{e}}{{\rm{c}}^2}\theta  – 1) – 31{\rm{cos}}{\rm{ec }}\theta  + {\rm{32}} = {\rm{0}}$

$12{\rm{cos}}{\rm{e}}{{\rm{c}}^2}\theta  – 31\,{\rm{cos}}{\rm{ec }}\theta  + {\rm{20}} = {\rm{0}}$

$12\,{\rm{cos}}{\rm{e}}{{\rm{c}}^2}\theta  – 16\,\,{\rm{cos}}{\rm{ec }}\theta  – 15\,{\rm{cos}}{\rm{ec}}\theta  + 20 = {\rm{0}}$

$(4\cos {\rm{ec}}\theta  – 5)(3\cos {\rm{ec}}\theta  – 4) = 0$

${\rm{cos}}{\rm{ec}}\theta  = \frac{5}{4},\frac{4}{3}$; 

$\therefore$  $\sin \theta  = \frac{4}{5},\frac{3}{4}$.