If $\mathrm{sin }\alpha =\frac{5}{13},\text{ then the value of }\cos \alpha \cdot \mathrm{cosec}\alpha \cdot \cot \alpha =\text{\_\_\_\_\_\_}.$

The correct answer is $\frac{144}{25}$.

We are given:
$\sin \alpha =\frac{5}{13}$
We need to calculate the value of:
$\cos \alpha \cdot \mathrm{cosec}\alpha \cdot \cot \alpha$

Step 1: Simplify the given expression using fundamental trigonometric ratios.
Recall the identities for cosecant and cotangent:
$\mathrm{cosec}\alpha =\frac{1}{\sin \alpha }$
$\cot \alpha =\frac{\cos \alpha }{\sin \alpha }$

Substitute these definitions into the expression:
$\cos \alpha \cdot \mathrm{cosec}\alpha \cdot \cot \alpha =\cos \alpha \cdot \frac{1}{\sin \alpha }\cdot \frac{\cos \alpha }{\sin \alpha }$
Multiplying the terms together gives:
$\cos \alpha \cdot \mathrm{cosec}\alpha \cdot \cot \alpha =\frac{{\cos }^2\alpha }{{\sin }^2\alpha }$

Step 2: Find ${\sin }^2\alpha$ and ${\cos }^2\alpha$.
Using the given value of $\sin \alpha =\frac{5}{13}$:
${\sin }^2\alpha ={\left(\frac{5}{13}\right)}^2=\frac{25}{169}$

Using the Pythagorean trigonometric identity ${\sin }^2\alpha +{\cos }^2\alpha =1$:
${\cos }^2\alpha =1-{\sin }^2\alpha$
${\cos }^2\alpha =1-\frac{25}{169}=\frac{169-25}{169}=\frac{144}{169}$

Step 3: Calculate the final simplified value.
Substitute ${\cos }^2\alpha$ and ${\sin }^2\alpha$ back into the ratio:
$\frac{{\cos }^2\alpha }{{\sin }^2\alpha }=\frac{\left(\frac{144}{169}\right)}{\left(\frac{25}{169}\right)}=\frac{144}{25}$

Hence, the value of $\cos \alpha \cdot \mathrm{cosec}\alpha \cdot \cot \alpha$ is $\frac{144}{25}$.