If the value of the trigonometric expression ${sec\theta + \tan \theta = x}$, then find the value of ${tan \theta}$

Hint: Use the trigonometric identity ${\sec ^2}\theta - {\tan ^2}\theta = 1$. Split it by using ${a^2}-{b^2}$ identity and proceed to find the value of ${tan \theta}$

Complete step by step answer:

Here we have

$\sec \theta + \tan \theta = x{\text{ }} -(1)$

By using trigonometric identity,

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

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

$\sec \theta - \tan \theta = \dfrac{1}{x}{\text{ }} -{\text{(2)}}$

Subtracting equation (2) from equation (1),we get,

$2\tan \theta = x - \dfrac{1}{x}$

$\tan \theta = \dfrac{1}{2}(x - \dfrac{1}{x})$

So, this is the required solution.

Note: In these types of questions we must carefully analyse which standard trigonometric equations are to be used. Also, we should have a grasp over trigonometric identities to solve the problems easily.