If $$\cosec \theta = 3x  and  \cot \theta = \frac{3}{x}$$, (x ≠ 0) then the value of $$6\left(x^2 - \frac{1}{x^2}\right)$$ is:

Given, $$\cosec\theta=3x$$ and $$\cot\theta=\frac{\ 3}{x}$$

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

$$=$$> $$\left(3x\right)^2-\left(\frac{\ 3}{x}\right)^2=1$$

$$=$$> $$9x^2-\frac{\ 9}{x^2}=1$$

$$=$$> $$9\left(x^2-\frac{\ 1}{x^2}\right)=1$$

$$=$$>  $$x^{2\ }-\frac{1}{x^2}=\frac{1}{9}$$

$$\therefore\ 6\left(x^{2\ }-\frac{1}{x^2}\right)=\frac{6}{9}=\frac{2}{3}$$

Hence, the correct answer is Option C