Question

If $\text{cosec}\theta =2x$ and $\cot \theta =\frac{2}{x},$ find the value of $2(x^2-\frac{1}{x^2})$.

Answer 1

Given:

$\text{cosec}\theta =2x,\cot \theta =\frac{2}{x}$

As we know that ${\text{cosec}}^2\theta -{\cot }^2\theta =1$

$\Rightarrow (2x{)}^2-(\frac{2}{x}{)}^2=1$

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

$\Rightarrow 4(x^2-\frac{1}{x^2})=1$

$\Rightarrow 2(x^2-\frac{1}{x^2})=\frac{1}{2}$

$\therefore 2(x^2-\frac{1}{x^2})=\frac{1}{2}$