Question

If (sin(()−1))p =2cos(()−1)q , show that p​^2​​ =4q​^2 (1− q^2) ?.

Answer 1

Likely the equation is

$\sin^{-1}p=2\cos^{-1}q$

For either side to be defined, we require $|p|\le1$ and $|q|\le1$. Under these conditions, we can take

$\sin(\sin^{1}p)=\sin(2\cos^{-1}q)$
$\implies p=2\sin(\cos^{-1}q)\cos(\cos^{-1}q)$
$\implies p=2q\sin(\cos^{-1}q)$
$\implies p=2q\sqrt{1-q^2}$
$\implies p^2=4q^2(1-q^2)$

as required.