# How do you prove sin(cos^-1x)=sqrt(1-x^2)?

$\sin \left({\cos}^{-} 1 x\right) = \sin \left({\sin}^{-} 1 \sqrt{1 - {x}^{2}}\right) = \sqrt{1 - {x}^{2}}$ (proved)
Let ${\cos}^{-} 1 x = \theta \therefore \cos \theta = x$. We know $\cos \theta = \frac{a}{h} = \frac{x}{1}$ where a(=x),adjacent side in a right angled triangle , and h(=1),is the hypotenuse, then by pythagorus theorm ,opposite side (o). $o = \sqrt{1 - {x}^{2}} \therefore \sin \theta = \frac{o}{h} = \frac{\sqrt{1 - {x}^{2}}}{1} \therefore \theta = {\sin}^{-} 1 \sqrt{1 - {x}^{2}}$
$\sin \left({\cos}^{-} 1 x\right) = \sin \theta = \sin \left({\sin}^{-} 1 \sqrt{1 - {x}^{2}}\right) = \sqrt{1 - {x}^{2}}$ (proved)