Exercise 2.11

Find expressions for (sin t/2)² and (cos t/2)² from the Pythagorean theorem and the cosine addition theorem.

Solution:

The Pythagorian theorem is the statement (sin x)² + (cos x)² = 1 (or we can interpret it to be that statement).

If we apply this equation for and also use the cosine addition theorem

cos (u + v) = (cos u)(cos v) - (sin u)(sin v)

for , we get

and we can add and subtract these equations from one another to get