Start from the right-hand side and rewrite in sine/cosine:
\[
\frac{\csc\theta\,\csc\varphi}{\cot\theta+\cot\varphi}
\;=\;
\frac{\dfrac{1}{\sin\theta}\cdot\dfrac{1}{\sin\varphi}}
{\dfrac{\cos\theta}{\sin\theta}+\dfrac{\cos\varphi}{\sin\varphi}}
\]
Combine the denominator over a common denominator:
\[
=\;
\frac{\dfrac{1}{\sin\theta\,\sin\varphi}}
{\dfrac{\cos\theta\,\sin\varphi+\cos\varphi\,\sin\theta}
{\sin\theta\,\sin\varphi}}
\]
Invert and multiply:
\[
=\;
\frac{1}{\cos\theta\,\sin\varphi+\cos\varphi\,\sin\theta}
\]
Recognize the sine addition formula
\(\sin(\theta+\varphi)=\sin\theta\cos\varphi+\cos\theta\sin\varphi\):
\[
=\;
\frac{1}{\sin(\theta+\varphi)}
\;=\;
\csc(\theta+\varphi).
\]
Hence \(\displaystyle
\csc(\theta+\varphi)\equiv
\dfrac{\csc\theta\,\csc\varphi}{\cot\theta+\cot\varphi}\).