Question

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

Answer 1

$$\begin{gathered} 2\left(x^2-\frac{1}{x^2}\right) \\ =\frac{4}{2}\left(x^2-\frac{1}{x^2}\right) \\ =\frac{1}{2}\left(4x^2-\frac{4}{x^2}\right) \\ =\frac{1}{2}\left[{\left(2x\right)}^2-{\left(\frac{2}{x}\right)}^2\right] \\ =\frac{1}{2}\left[{\left(\mathrm{cosec}\theta \right)}^2-{\left(\sec \theta \right)}^2\right] \\ =\frac{1}{2}\left({\mathrm{cosec}}^2\theta -{\sec }^2\theta \right) \\ =\frac{1}{2}\left(1\right) \\ =\frac{1}{2} \end{gathered}$$