Question

Find $x$ if ${\sin }^{-1}(1-x)-2{\sin }^{-1}x=\frac{\pi }{2}$.

Answer 1

Let ${\sin }^{-1}(1-x)=A$ so $\sin A=1-x$

Let ${\sin }^{-1}\left(x\right)=B$ so $\sin B=x$

$A-2B=\pi /2$

$A-\pi /2=2B$

$\cos (A-\pi /2=\cos 2B)$

$\sin A=1-2{\sin }^{2}B$

$1-x=1-2x^2$

$2x^2-x=0$

$x=0x=1/2$