Question

If $\sec \theta =x+\frac{1}{4x}$ then prove that $\sec \theta +\tan \theta =2x$ or, $\frac{1}{2x}$

Answer 1

$\sec \theta =x+\frac{1}{4x}$
Lets take $\sec \theta +\tan \theta =p\to 1$
($\because$same value p)
$\Rightarrow (\sec \theta +\tan \theta )\times \frac{\sec \theta -\tan \theta }{\sec \theta -\tan \theta }=p$
$\Rightarrow \frac{{\sec }^2\theta -{\tan }^2\theta }{\sec \theta -\tan \theta }=p$
$\Rightarrow \frac{1}{\sec \theta -\tan \theta }=p$
$\Rightarrow \sec \theta -\tan \theta =\frac{1}{p}\to 2$
$1+2$
$2\sec \theta =p+\frac{1}{p}$We know that $\sec \theta =x+\frac{1}{4x}$
$2(x+\frac{1}{4x})=p+\frac{1}{p}$
So $p=2x$ or
$\frac{1}{2x}$ (both values are satisfied)