Answer
See the proof below.
Work Step by Step
\begin{align*}
\lim _{\theta \rightarrow 0} \csc \theta -\cot\theta &= \lim _{\theta \rightarrow 0} \frac{1}{\sin \theta}-\frac{\cos \theta}{\sin \theta}\\
&=\lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{\sin \theta}\\
&= \lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{\theta} \frac{\theta}{\sin \theta}\\
&= \lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{\theta} \lim _{\theta \rightarrow 0} \frac{\theta}{\sin \theta}\\
&=0.
\end{align*}
Where we used Theorem 2 -- that is, $\lim _{x\rightarrow 0}\frac{ \sin x}{ x}=1$ and $\lim _{x\rightarrow 0}\frac{ 1-\cos x}{x}=0. $
