Question

Evaluate:

\[x^2 - \frac{1}{x^2}\] when \[\operatorname{cosec} A = 2x\] and \[\tan A = \frac{x}{2}\]

Answer 1

Given,

\[\operatorname{cosec} A = 2x\] and \[\tan A = \frac{x}{2}\]

\[\Rightarrow \cot A = \frac{1}{\tan A} = \frac{2}{x}\]

By formula,

\[\operatorname{cosec}^2 A - \cot^2 A = 1\]

\[\Rightarrow (2x)^2 - \left(\frac{2}{x}\right)^2 = 1\]

\[\Rightarrow 4x^2 - \frac{4}{x^2} = 1\]

\[\Rightarrow 4\left(x^2 - \frac{1}{x^2}\right) = 1\]

\[\Rightarrow x^2 - \frac{1}{x^2} = \frac{1}{4}\]

Hence, \[x^2 - \frac{1}{x^2} = \frac{1}{4}\]