The number of solutions of the equation, ${sin}^{- 1} x = 2 {tan}^{- 1} x$ (in principle value) is:-

3

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Solution

The correct option is D $3$
$s i n^{- 1} x = 2 t a n^{- 1} x$

$s i n^{- 1} x = s i n^{- 1} (\frac{2 x}{1 + x^{2}})$

$x = \frac{2 x}{1 + x^{2}}$

$x (1 + x^{2}) = 2 x$

$x + x^{3} = 2 x$

$x^{3} - x = 0$

Hence $0, - 1, 1$ are

solution of equation
i.e. $x (x^{2} - 1^{2}) = 0$

$x (x + 1) (x - 1) = 0$

$x = 0, x + 1 = 0, x - 1 = 0$

$x = - 1, x = 1$

$∴$ It has 3 solutions.

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