#### Need Solution for R.D.Sharma Maths Class 12 Chapter 3 Inverse Trigonomeric Functions Exercise Multiple Choice Questions  Question 3 Maths Textbook Solution.

Answer: $\tan ^{-1} x$

Hint: Using trigonometric functions, try to solve.

Given: $2 \tan ^{-1}\left\{\csc \left(\tan ^{-1} x\right)-\tan \left(\cot ^{-1} x\right)\right\}$

Solution:

Let,$\tan ^{-1} x=y$

So, $\mathrm{x}=\tan \mathrm{y}$

$\therefore 2 \tan ^{-1}\left\{\csc \left(\tan ^{-1} x\right)-\tan \left(\cot ^{-1} x\right)\right\}=2 \tan ^{-1}\left\{\csc \left(\tan ^{-1} x\right)-\tan \left(\tan ^{-1} \frac{1}{x}\right)\right\}$

$=2 \tan ^{-1}\left\{\csc \left(\tan ^{-1} x\right)-\frac{1}{x}\right\}$

$=2 \tan ^{-1}\left\{\csc y-\frac{1}{\tan y}\right\}$

$=2 \tan ^{-1}\left\{\frac{1-\cos y}{1-\sin y}\right\}$

$=2 \tan ^{-1}\left\{\frac{2 \sin ^{2} \frac{y}{2}}{\sin y}\right\}$

$=2 \tan ^{-1}\left\{\frac{2 \sin ^{2} \frac{y}{2}}{2 \sin \frac{y}{2} \cos \frac{y}{2}}\right\}$

$=2 \tan ^{-1}\left\{\tan \frac{y}{2}\right\}$

$=\mathrm{y}$         Alignment of the solution needs to be done properly

$=\tan ^{-1} x$