$\text{ If }\mathrm{f}\left(\mathrm{x}\right)={\int }_0^{{\sin }^2\mathrm{x}}{\sin }^{-1}\sqrt{\mathrm{t}}\mathrm{dt}+{\int }_0^{{\cos }^2\mathrm{x}}{\cos }^{-1}\sqrt{\mathrm{t}}\mathrm{dt},\text{ then which of the following is not true? }$

$\begin{array}{l}{f}^1\left(x\right)={\sin }^{-1}\left(\sin x\right)2\sin x\cos x+{\cos }^{-1}\left(\cos x\right)\left(-2\sin x\cos x\right)=0\\ \Rightarrow f\left(x\right)=\text{ constant }\\ \Rightarrow f\left(x\right)=f\left(\frac{\pi }{4}\right)={\int }_0^{\frac{1}{2}}{\sin }^{-1}\left(\sqrt{t}\right)dt+{\int }_0^{\frac{1}{2}}{\cos }^{-1}\left(\sqrt{t}\right)dt\\ ={\int }_0^{\frac{1}{2}}\left[s{\mathrm{in}}^{-1}\left(\sqrt{t}\right)+{\cos }^{-1}\left(\sqrt{t}\right)\right]dt\\ =\frac{\mathrm{\pi }}{2}{\int }_0^{\frac{1}{2}}dt\\ =\frac{\mathrm{\pi }}{4}\\ \therefore f\left(x\right)=\frac{\mathrm{\pi }}{4}\forall x\in R\\ Now f\left(\frac{\mathrm{\pi }}{4}\right)+f\left(\frac{\mathrm{\pi }}{2}\right)=\frac{\mathrm{\pi }}{2}\\ \end{array}$