Question:
If $3 x=\operatorname{cosec} \theta$ and $\frac{3}{x}=\cot \theta$, find the value of $3\left(x^{2}-\frac{1}{x^{2}}\right)$.
Solution:
$3\left(x^{2}-\frac{1}{x^{2}}\right)$
$=\frac{9}{3}\left(x^{2}-\frac{1}{x^{2}}\right)$
$=\frac{1}{3}\left(9 x^{2}-\frac{9}{x^{2}}\right)$
$=\frac{1}{3}\left[(3 x)^{2}-\left(\frac{3}{x}\right)^{2}\right]$
$=\frac{1}{3}\left[(\operatorname{cosec} \theta)^{2}-(\cot \theta)^{2}\right]$
$=\frac{1}{3}\left(\operatorname{cosec}^{2} \theta-\cot ^{2} \theta\right)$
$=\frac{1}{3}(1)$
$=\frac{1}{3}$
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