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The solution set of the equation \sin ^{-1} x=2 \tan ^{-1} x is

Option: 1

{\{1,2\}}


Option: 2

{\{-1,2\}}


Option: 3

{\{-1,1,0\}}


Option: 4

{\{1,1 / 2,0\}}


Answers (1)

best_answer

 

 

Multiple angles in terms of  arctan and arcsin -

Multiple angles in terms of  arctan and arcsin 

 

 

\\\mathrm{\;\;\;2\;\tan^{-1}x=\left\{\begin{matrix} \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right), & & \text { if }-1 \leq x \leq 1\\ \\ \pi-\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right),& &\text { if }\;\;x>1 \\\\ -\pi-\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right),& & \text { if }\;\;x<-1 \end{matrix}\right.}

Now,

\sin ^{-1} x=2 \tan ^{-1} x\\ \sin ^{-1} x=\sin ^{-1} \frac{2x}{1+x^2}\\ x=\frac{2x}{1+x^2}\\ x=0, x^2+1=2\rightarrow x=\pm 1

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Divya Prakash Singh

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