]> Exercise 2.11

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 x = t 2 and also use the cosine addition theorem

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

for u = v = t 2 , we get

cos t = cos 2 t 2 sin 2 t 2 1 = cos 2 t 2 + sin 2 t 2

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

sin 2 t 2 = 1 cos t 2 cos 2 t 2 = 1 + cos t 2