${\text{The number of values of x for which sin}}^{-1}\left(x^2-\frac{x^4}{3}+\frac{x^6}{9}......\right){\text{+cos}}^{-1}\left(x^4-\frac{x^8}{3}+\frac{x^{12}}{9}......\right)\text{=}\frac{\pi }{2}\text{,0≤}\left|x\right|\text{<}\sqrt{3}\text{,is}$

${\sin }^{-1}\left(x^2-\frac{x^4}{3}+\frac{x^6}{9}......\right){\text{+cos}}^{-1}\left(x^4-\frac{x^8}{3}+\frac{x^{12}}{9}......\right)\text{=}\frac{\pi }{2}$

$\Rightarrow \left(x^2-\frac{x^4}{3}+\frac{x^6}{9}......\right)=\left(x^4-\frac{x^8}{3}+\frac{x^{12}}{9}......\right)$      $\left(\sin ce {\sin }^{-1}\left(x\right)+{\cos }^{-1}\left(x\right)=\frac{\mathrm{\pi }}{2}\right)$

$\Rightarrow \frac{x^2}{1+{\displaystyle \frac{x^2}{3}}}=\frac{x^4}{1+{\displaystyle \frac{x^4}{3}}}$       $\left(sum of n terms in G.P = \frac{a}{1-r}\right)$

$\Rightarrow x^2=0 or \frac{3}{3+x^2}=\frac{3x^2}{3+x^4}$

$\Rightarrow x^2=1$

$\Rightarrow x=0,1 \text{or -1}$

$\text{number of values of x is equal to 3}$