MHB Evaluating cos((1/2)arccos(x))

[Elissa89]

I don't know how to solve this, we didn't really cover any problems like this in class

cos(1/2*cos^-1*x)

This is due tonight online and would like help please.

 

[MarkFL]

Re: Need help, due tonight

The first thing I would consider is:

$$0\le\arccos(x)\le\pi$$

Hence:

$$0\le\frac{1}{2}\arccos(x)\le\frac{\pi}{2}$$

This means the cosine of the given angle will be non-negative. Next, consider the half-angle identity for cosine:

$$\cos^2\left(\frac{\theta}{2}\right)=\frac{1+\cos(\theta)}{2}$$

Given that the cosine function will be non-negative, we may write:

$$\cos\left(\frac{\theta}{2}\right)=\sqrt{\frac{1+\cos(\theta)}{2}}$$

Can you proceed?