# How do you prove (cos[theta]+cot[theta])/(csc[theta]+1)= cos[theta]?

Using the definitions of $\cot \left(\theta\right)$ and $\csc \left(\theta\right)$, for $\sin \left(\theta\right) \ne 0$ and $\sin \left(\theta\right) \ne - 1$, we have
$\frac{\cos \left(\theta\right) + \cot \left(\theta\right)}{\csc \left(\theta\right) + 1} = \frac{\cos \left(\theta\right) + \cos \frac{\theta}{\sin} \left(\theta\right)}{\frac{1}{\sin} \left(\theta\right) + 1}$
$= \cos \left(\theta\right) \frac{1 + \frac{1}{\sin} \left(\theta\right)}{1 + \frac{1}{\sin} \left(\theta\right)}$
$= \cos \left(\theta\right) \cdot 1$
$= \cos \left(\theta\right)$