The number of roots of the equation ${\sin }^{-1}\frac{3x}{5}+{\sin }^{-1}\frac{4x}{5}={\sin }^{-1}x$ is

${\sin }^{-1}\frac{3x}{5}+{\sin }^{-1}\frac{4x}{5}={\sin }^{-1}x$

$\Rightarrow {\sin }^{-1}(\frac{3x}{5}\sqrt{1-\frac{16x^2}{25}}+\frac{4x}{5}\sqrt{1-\frac{9x^2}{25}})={\sin }^{-1}x$

$\Rightarrow \frac{3x}{5}\frac{\sqrt{25-16x^2}}{5}+\frac{4x}{5}\frac{\sqrt{25-9x^2}}{5}=x$

$\Rightarrow x=0or3\sqrt{25-16x^2}+4\sqrt{25-9x^2}=25\to \left(1\right)$

$\Rightarrow \frac{175}{\sqrt{400-144x^2}-\sqrt{225-144x^2}}=25$

$\Rightarrow \sqrt{400-144x^2}-\sqrt{225-144x^2}=7\to \left(2\right)$

From (1) and (2),

$\sqrt{400-144x^2}=16$

$\Rightarrow 400-144x^2=256$

$\Rightarrow x^2=1$

$\Rightarrow x=\pm 1$