How to use the half angle of cosine when given a triangle

The video tutorial explains how to find the cosine of Theta over 2 for a triangle using the formula sqrt(1 + cos(Theta)/2). It covers the importance of understanding when to use positive or negative square roots based on the angle's position. The tutorial demonstrates using the Pythagorean theorem to find the hypotenuse and evaluates the cosine of Theta. It also includes steps to simplify fractions and rationalize the denominator to reach the final answer.

Explain how the Pythagorean theorem is applied in the context of finding the hypotenuse in the given example.

What steps are taken to rationalize the denominator in the final answer?