]> 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

$sin ⁡ 2 t 2 = 1 − cos ⁡ t 2$ $cos ⁡ 2 t 2 = 1 + cos ⁡ t 2$