Question

Prove that :
${\cos }^{-1}x=2{\sin }^{-1}\left(\sqrt{\frac{1-x}{2}}\right).$

Answer 1

R.E.F image

Solution -

$co{s}^{-1}x=2si{n}^{-1}\sqrt{\frac{1-x}{2}}$

R.H.S

Let $y=2si{n}^{-1}\sqrt{\frac{1-x}{2}}$

$\frac{y}{2}=co{s}^{-1}\left(\frac{\sqrt{1+x}}{\sqrt{2}}\right)$

$co{s}^2\left(\frac{y}{2}\right)=\frac{1+x}{2}$

$cosy=2co{s}^2\frac{y}{2}-1=1+x-1$

$cosy=x$

$y=co{s}^{-1}x$

$=LHS$