Exercise 2.11

]> Exercise 2.11

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Find expressions for ${(\sin \frac{t}{2})}^{2}$ and ${(\cos \frac{t}{2})}^{2}$ 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 $x = \frac{t}{2}$ and also use the cosine addition theorem

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

for $u = v = \frac{t}{2}$ , we get

\cos t = \cos^{2} \frac{t}{2} - \sin^{2} \frac{t}{2}

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

\sin^{2} \frac{t}{2} = \frac{1 - \cos t}{2}