Answer
$\csc(\theta)+\sin(-\theta)=\frac{\cos^{2}(\theta)}{\sin(\theta)}$
Work Step by Step
$\csc(\theta)$ can also be represented as $\frac{1}{\sin(\theta)}$
Since sine is an odd function, $\sin(-\theta)=-\sin(\theta)$
The equation can be rewritten as:
$\csc(\theta)+\sin(-\theta)=\frac{1}{\sin(\theta)}-\sin(\theta)$
The denominators of both terms can now be converted to $\sin(\theta)$ by multiplying $\sin(\theta)$ by $\frac{\sin(\theta)}{\sin(\theta)}$ to give $\frac{\sin^{2}(\theta)}{\sin(\theta)}$.
As such,
$\frac{1}{\sin(\theta)}-\sin(\theta)=\frac{1-\sin^{2}(\theta)}{\sin(\theta)}$
By using the identity $1-\sin^{2}(\theta)=\cos^{2}(\theta)$, the equation can be simplified to:
$\frac{1-\sin^{2}(\theta)}{\sin(\theta)}=\frac{\cos^{2}(\theta)}{\sin(\theta)}$
Therefore,
$\csc(\theta)+\sin(-\theta)=\frac{\cos^{2}(\theta)}{\sin(\theta)}$
