Question

If $3x=\mathrm{cosec}\theta$ and $\frac{3}{x}=\cot \theta$, find the value of $3\left(x^2-\frac{1}{x^2}\right)$. [CBSE 2010]

Answer 1

$$\begin{gathered} 3\left(x^2-\frac{1}{x^2}\right) \\ =\frac{9}{3}\left(x^2-\frac{1}{x^2}\right) \\ =\frac{1}{3}\left(9x^2-\frac{9}{x^2}\right) \\ =\frac{1}{3}\left[{\left(3x\right)}^2-{\left(\frac{3}{x}\right)}^2\right] \end{gathered}$$
$$\begin{gathered} =\frac{1}{3}\left[{\left(\mathrm{cosec}\theta \right)}^2-{\left(\cot \theta \right)}^2\right] \\ =\frac{1}{3}\left({\mathrm{cosec}}^2\theta -{\cot }^2\theta \right) \\ =\frac{1}{3}\left(1\right) \\ =\frac{1}{3} \end{gathered}$$