If $\sin \alpha =\frac{5}{13}$, then the value of $\cos \alpha \cdot \csc \alpha \cdot \cot \alpha$ is _________.

Given:
$$\sin \alpha =\frac{5}{13}$$

We need to find the value of $\cos \alpha \cdot \csc \alpha \cdot \cot \alpha$.

Let us simplify the given expression using basic trigonometric identities:

$$\cos \alpha \cdot \csc \alpha \cdot \cot \alpha =\cos \alpha \cdot \frac{1}{\sin \alpha }\cdot \frac{\cos \alpha }{\sin \alpha }$$

$$=\frac{{\cos }^2\alpha }{{\sin }^2\alpha }={\cot }^2\alpha$$

Using the identity ${\cos }^2\alpha =1-{\sin }^2\alpha$:

$${\cos }^2\alpha =1-{\left(\frac{5}{13}\right)}^2=1-\frac{25}{169}=\frac{144}{169}$$

Now, substitute the values of ${\cos }^2\alpha$ and ${\sin }^2\alpha$:

$${\cot }^2\alpha =\frac{{\cos }^2\alpha }{{\sin }^2\alpha }=\frac{\frac{144}{169}}{\frac{25}{169}}=\frac{144}{25}$$

Therefore, the value of the expression is $\frac{144}{25}$.