# If cosec θ = 2x and cot θ=2x, find the value of 2(x2−1x2).

Question:

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

Solution:

Given:

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

We know that,

$\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1$

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

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

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

$\Rightarrow \quad 2 \times 2 \times\left(x^{2}-\frac{1}{x^{2}}\right)=1$

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