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

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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\}


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



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