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,
$\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1$
$\Rightarrow(2 x)^{2}-\left(\frac{2}{x}\right)^{2}=1$
$\Rightarrow \quad 4 x^{2}-\frac{4}{x^{2}}=1$
$\Rightarrow 4\left(x^{2}-\frac{1}{x^{2}}\right)=1$
$\Rightarrow \quad 2 \times 2 \times\left(x^{2}-\frac{1}{x^{2}}\right)=1$
$\Rightarrow \quad 2\left(x^{2}-\frac{1}{x^{2}}\right)=\frac{1}{2}$
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