$\ {\sin}^{-1}{(x)} $ equation and function

Question

$$\ {F(x)} = {\sin}^{-1}{(2x-1)} - 2{\sin}^{-1}{\sqrt x} $$ I know that $\ F(x) $ is constant function and equal to $\ -\frac{\pi}{2} $ on its domain .
First how can I understand that without checking different numbers?
Second how to prove it?

Answer 1

we get by differentating with respect to $x$ $$F'(x)={\frac {1}{\sqrt {-{x}^{2}+x}}}-{\frac {1}{\sqrt {x}\sqrt {-x+1}}}$$ Can you finish?