If $$\cot \theta = 5x  and  \cosec \theta = \frac{5}{x}$$(x ≠ 0), then the value of $$5\left(x^2 - \frac{1}{x^2}\right)$$ is:

Given, $$\cot\theta\ =5x$$ and $$\operatorname{cosec}\theta\ =\frac{5}{x}$$

We know that, $$\operatorname{cosec}^2\theta\ -\cot^2\theta=1\ $$

$$=$$>  $$\left(\frac{5}{x}\right)^2-\left(5x\right)^2=1\ $$

$$=$$>  $$\frac{25}{x^2}-25x^2=1\ $$

$$=$$>  $$25\left(\frac{1}{x^2}-x^2\right)=1\ $$

$$=$$>  $$x^2-\frac{1}{x^2}=\frac{-1}{25}\ $$

$$=$$>  $$5\left(x^2-\frac{1}{x^2}\right)=\frac{-5}{25}\ $$

$$=$$>  $$5\left(x^2-\frac{1}{x^2}\right)=\frac{-1}{5}\ $$

Hence, the correct answer is Option A