If $\mathrm{cosec}\theta =\frac{5}{3}$, then evaluate $({\sec }^2\theta -1)\times {\cot }^2\theta \times (1+{\cot }^2\theta )$

The correct option is $\frac{25}{9}$.

To evaluate the expression, we start with the given value:
$\mathrm{cosec}\theta =\frac{5}{3}$

Let us first simplify the expression we want to evaluate:
$E=({\sec }^2\theta -1)\times {\cot }^2\theta \times (1+{\cot }^2\theta )$

Using standard trigonometric identities:
1. ${\sec }^2\theta -1={\tan }^2\theta$
2. $1+{\cot }^2\theta ={\mathrm{cosec}}^2\theta$

Substituting these identities into the expression:
$E={\tan }^2\theta \times {\cot }^2\theta \times {\mathrm{cosec}}^2\theta$

Since tangent and cotangent are reciprocals of each other, we have:
${\tan }^2\theta \times {\cot }^2\theta =1$

Therefore, the expression simplifies to:
$E=1\times {\mathrm{cosec}}^2\theta ={\mathrm{cosec}}^2\theta$

Now, substitute the given value of $\mathrm{cosec}\theta$:
$E={\left(\frac{5}{3}\right)}^2=\frac{25}{9}$