Answer
The Derivative is:
$\frac{dr}{d\theta}=\cos\theta+\sec^2\theta$
Work Step by Step
$r=(1+\sec\theta)\sin\theta$
Using the distributive property of mathematics
$r=\sin\theta+\sec\theta\sin\theta$
$r=\sin\theta+\frac{1}{\cos\theta}\cdot\sin\theta$
$r=\sin\theta+\tan\theta$
Applying Derivative rules:
$y'=f'(x)+g'(x)$
$\frac{dr}{d\theta}=\frac{d}{d\theta}(\sin\theta)+\frac{d}{d\theta}(\tan\theta)$
$\frac{dr}{d\theta}=(\cos\theta)+(\sec^2\theta)$
$\frac{dr}{d\theta}=\cos\theta+\sec^2\theta$
