Answer
$\frac{dr}{d\theta} = \frac{(cosec \theta)}{(cosec \theta +cot \theta)}$
Work Step by Step
$r = u^{-1}$
where,
$u = (cosec \theta +cot \theta)$
Now,
$\frac{dr}{du} = -u^{-2}$
and,
$\frac{du}{d \theta} = -cosec \theta \times cot \theta - cosec ^{2} \theta = -cosec \theta(cot \theta + cosec \theta)$
So,
$\frac{dr}{d \theta} = \frac{dr}{du} \times \frac{du}{d\theta}$
$\frac{dr}{d\theta} = -u^{-2}(-cosec \theta(cot \theta + cosec \theta))$
$\frac{dr}{d\theta} = (cosec \theta +cot \theta)^{-2}(cosec \theta(cot \theta + cosec \theta))$
$\frac{dr}{d\theta} = \frac{(cosec \theta(cot \theta + cosec \theta))}{(cosec \theta +cot \theta)^{2}}$
$\frac{dr}{d\theta} = \frac{(cosec \theta)}{(cosec \theta +cot \theta)}$