Question

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

Answer 1

Given,

$\tan \theta +\sec \theta =2x$......(1).

We've,

${\sec }^2\theta -{\tan }^2\theta =1$

or, $(\sec \theta -\tan \theta )(\sec \theta +\tan \theta )=1$

or, $\sec \theta -\tan \theta =\frac{1}{2x}$......(2) [ Using (1)].

Now, adding (1) and (2) we get,

$2\sec \theta =2x+\frac{1}{2x}$

or, $\sec \theta =x+\frac{1}{4x}$.