Question

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

• A. 0
• B. 1
• C. 2
• D. infinite

Answer 1

Answer: (D)

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.