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If $12{\cot ^2}\theta - 31\,{\rm{cosec }}\theta + {\rm{32}} = {\rm{0}}$, then the value of $\sin \theta $ is
$\frac{3}{5}$ or $1$
$\frac{{2}}{3}$ or $\frac{{ - 2}}{3}$
$\frac{4}{5}$ or $\frac{3}{4}$
$ \pm \frac{1}{2}$
Solution
(c) $12{\cot ^2}\theta – 31\cos ec\theta + 32 = 0$
$12({\rm{cos}}{\rm{e}}{{\rm{c}}^2}\theta – 1) – 31{\rm{cos}}{\rm{ec }}\theta + {\rm{32}} = {\rm{0}}$
$12{\rm{cos}}{\rm{e}}{{\rm{c}}^2}\theta – 31\,{\rm{cos}}{\rm{ec }}\theta + {\rm{20}} = {\rm{0}}$
$12\,{\rm{cos}}{\rm{e}}{{\rm{c}}^2}\theta – 16\,\,{\rm{cos}}{\rm{ec }}\theta – 15{\rm{cos}}{\rm{ec}}\theta + 20 = {\rm{0}}$
$(4\cos {\rm{ec}}\theta – 5)(3\cos {\rm{ec}}\theta – 4) = 0$
${\rm{cos}}{\rm{ec}}\theta = \frac{5}{4},\frac{4}{3}$;
$\therefore$ $\sin \theta = \frac{4}{5},\frac{3}{4}$.
