Question

$\mathrm{If}\sin \theta =\frac{1}{2},\mathrm{then}\mathrm{write}\mathrm{the}\mathrm{value}\mathrm{of}\left(3{\cot }^2\theta +3\right).\left[\mathrm{CBSE}2009\right]$

Answer 1

$$\begin{gathered} \mathrm{As},\sin \theta =\frac{1}{2} \\ \mathrm{So},\mathrm{cosec}\theta =\frac{1}{\sin \theta }=2.....\left(\mathrm{i}\right) \end{gathered}$$

Now,

$$\begin{gathered} 3{\cot }^2\theta +3 \\ =3\left({\cot }^2\theta +1\right) \\ =3{\mathrm{cosec}}^2\theta \\ =3{\left(2\right)}^2\left[\mathrm{Using}\left(\mathrm{i}\right)\right] \\ =3\left(4\right) \\ =12 \end{gathered}$$