If sin θ=13, then find the value of 2cot2 θ + 2.

Given:

$\sin \theta=\frac{1}{3}$

$\Rightarrow \frac{1}{\sin \theta}=3$

$\Rightarrow \operatorname{cosec} \theta=3$

We know that,

$\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1$

$\Rightarrow(3)^{2}-\cot ^{2} \theta=1$

$\Rightarrow \cot ^{2} \theta=9-1$

$\Rightarrow \cot ^{2} \theta=8$

Therefore,

$2 \cot ^{2} \theta+2=2 \times 8+2$

$=16+2$

$=18$