Likely the equation is
[tex]\sin^{-1}p=2\cos^{-1}q[/tex]
For either side to be defined, we require [tex]|p|\le1[/tex] and [tex]|q|\le1[/tex]. Under these conditions, we can take
[tex]\sin(\sin^{1}p)=\sin(2\cos^{-1}q)[/tex]
[tex]\implies p=2\sin(\cos^{-1}q)\cos(\cos^{-1}q)[/tex]
[tex]\implies p=2q\sin(\cos^{-1}q)[/tex]
[tex]\implies p=2q\sqrt{1-q^2}[/tex]
[tex]\implies p^2=4q^2(1-q^2)[/tex]
as required.
