How to simplify of $\sin\left(\frac{\cos^{-1}\theta}2\right)$?

Question

Is there any way to remove the $\cos^{-1}$ from $\sin\left(\dfrac{\cos^{-1}\theta}2\right)$?

Answer 1

Use the half-angle formula: $$\sin\frac A2=\pm\sqrt{\frac{1-\cos A}2}$$ to get $$\sin\left(\frac{\cos^{-1}\theta}2\right)=\pm\sqrt{\frac{1-\theta}2}$$

Answer 2

We have the following identity: $$\sin^2(x/2)=\frac{1-\cos(x)}{2}$$

So if you set $x=\cos^{-1}(θ)$: $$\sin^2( \cos^{-1}(θ) /2)=\frac{1-\cos( \cos^{-1} (θ) )}{2}= \frac{1-θ }{2} $$ And from there you can get what you want.