Answer
$\cot\theta$ = $-\frac{\sqrt {1-\sin^{2}\theta}}{\sin\theta}$
($\sin\theta\ne0$ as $\theta$ lies in quadrant II)
Work Step by Step
We are supposed to write $\cot\theta$ in terms of $\sin\theta$ while $\theta$ lies in Quadrant II.
Using ratio identity for $\cot$-
$\cot\theta$ = $\frac{\cos\theta}{\sin\theta}$
($\sin\theta\ne0$ as $\theta$ lies in quadrant II)
From Pythagorean identity-
$\cos\theta$ may be written as $\sqrt {1-\sin^{2}\theta}$
Therefore-
$\cot\theta$ = $-\frac{\sqrt {1-\sin^{2}\theta}}{\sin\theta}$
($\sin\theta\ne0$ as $\theta$ lies in quadrant II)
Since the angle is in quadrant II, where cotangent is negative but sine is positive. The negative value of the square root is chosen to make cotangent negative.
