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  4. Number of solutions satisfying the equation sin -1((2 x/1+x2))+ cos -1((1-x2/1+x2))=0 is

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Q. Number of solutions satisfying the equation $\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)+\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)=0$ is

Inverse Trigonometric Functions

Solution:

image
If $x \in[0,1] \Rightarrow 2 \tan ^{-1} x+2 \tan ^{-1} x=0$
$\therefore x =0$
If $x \in[-1,0] \Rightarrow 2 \tan ^{-1} x-2 \tan ^{-1} x=0 \Rightarrow$ Identity
If $x \in(-\infty,-1] \Rightarrow-\pi-2 \tan ^{-1} x-2 \tan ^{-1} x=0$
$4 \tan ^{-1} x=\frac{-\pi}{4} $
$x=-1$
$\therefore$ Infinite solution.