Question:
If $5 x=\sec \theta$ and $\frac{5}{x}=\tan \theta$, find the value of $5\left(x^{2}-\frac{1}{x^{2}}\right)$
Solution:
Given:
$5 x=\sec \theta, \frac{5}{x}=\tan \theta$
$\Rightarrow \sec \theta=5 x, \tan \theta=\frac{5}{x}$
We know that,
$\sec ^{2} \theta-\tan ^{2} \theta=1$
$\Rightarrow(5 x)^{2}-\left(\frac{5}{x}\right)^{2}=1$
$\Rightarrow 25 x^{2}-\frac{25}{x^{2}}=1$
$\Rightarrow 5 \times 5 \times\left(x^{2}-\frac{1}{x^{2}}\right)=1$
$\Rightarrow 5\left(x^{2}-\frac{1}{x^{2}}\right)=\frac{1}{5}$