Find the exact value of cot(15degrees) using half angle formulas?

Double angle (or half angle, depending how you look at it) identities:

So we have

Note that when taking the square root, we should take into account that that could yield two possible solutions, but we know and , so it's also the case that .

Also, the reason we have equality in the last step can be explained like so:

(not unlike the process used to complete the square)

could you provide figure.