The number of solutions of the equation ${\sin }^{-1}\left[x^2+\frac{1}{3}\right]+{\cos }^{-1}\left[x^2-\frac{2}{3}\right]=x^2$ , for $x\in \left[-1,1\right]$ , and $\left[x\right]$ denotes the greatest integer less than or equal to $x$ , is:

$\begin{array}{l}{\sin }^{-1}\left[x^2+\frac{1}{3}\right]+{\cos }^{-1}\left(\left[x^2+\frac{1}{3}\right]-1\right)=x^2\\ {\sin }^{-1}t+{\cos }^{-1}\left(t-1\right)=x^2\text{\hspace{0.17em} where }t=\left[x^2+\frac{1}{3}\right]; x\in \left[-1,1\right]\Rightarrow t=0,1\\ t=0\Rightarrow x^2=\pi \&x^2<\frac{2}{3} this is not possible\\ and when t =1, then x^2=\mathrm{\pi } \mathrm{and} \frac{2}{3}<{\mathrm{x}}^2<\frac{5}{3}, \mathrm{this} \mathrm{also} \mathrm{not} \mathrm{possible}. \end{array}$