# Question #13d24

Nov 17, 2016

see below

#### Explanation:

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

We will prove this by using the double argument formula for cosine.

Recall that one of the formula for $\cos 2 A$ is $\cos 2 A = 2 {\cos}^{2} A - 1$

So if we isolate cos A then we have

$\frac{\cos 2 A + 1}{2} = {\cos}^{2} A$

$\pm \sqrt{\frac{1 + \cos 2 A}{2}} = \cos A$

So from this formula we can see that you double angle A inside the square root. Since A in this case is $\frac{1}{2} A$ then we have $2 A = 2 \left(\frac{1}{2} A\right) = A$,

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

$\therefore \pm \sqrt{\frac{1 + \cos A}{2}} = \cos \left(\frac{A}{2}\right)$