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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

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