Question

How do you find the value of $\tan \left( {{{\tan }^{ - 1}}\left( {\dfrac{1}{2}} \right)} \right)$?

Answer 1

Hint : First of all observe the given mathematical expression, and apply the basic standard identity to get its simplified equivalent value. Here, we are given trigonometric tangent function and its inverse function and so here our solution becomes simpler being the same trigonometric function provided it falls under the domain values.

Complete step-by-step answer :
Take the given expression: $\tan \left( {{{\tan }^{ - 1}}\left( {\dfrac{1}{2}} \right)} \right)$
The inverse trigonometric functions are also known as arcus functions, anti-trigonometric functions or the cyclometric functions.
In simple terms, the inverse trigonometric functions are the inverse functions of the trigonometric functions.
Tangent and the inverse of the tangent are opposite to each other and they cancel each other. By using the identity $\tan ({\tan ^{ - 1}}\theta ) = \theta$
So, here
$\tan \left( {{{\tan }^{ - 1}}\left( {\dfrac{1}{2}} \right)} \right) = \left( {\dfrac{1}{2}} \right)$
This is the required solution.
So, the correct answer is “$\dfrac{1}{2}$”.

Note : Always remember that the inverse of the tangent function with the tangent function cancels each other. Same trigonometric function and its inverse cancel each other out. Always remember the formulas for the integration of trigonometric functions and its value for efficiency and the correct solution and also be careful about the sign convention.