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 \(\displaystyle \csc(\theta+\varphi)\equiv \dfrac{\csc\theta\,\csc\varphi}{\cot\theta+\cot\varphi}\).