Proof of the bisection formulae

The bisection formulae are nothing but the cosine duplication formulae “read backwards”, with the variable $\alpha/2$ in place of $\alpha$.

Proof — Bisection formulae

From the duplication formula $\cos\alpha = 1-2\sin^2(\alpha/2)$ we obtain $\sin^2\!\left(\frac{\alpha}{2}\right) = \frac{1-\cos\alpha}{2},$ from which, taking the square root, we get the formula for $\sin(\alpha/2)$ with the sign to be determined according to the quadrant. Analogously, from the form $\cos\alpha = 2\cos^2(\alpha/2) - 1$ we obtain $\cos^2\!\left(\frac{\alpha}{2}\right) = \frac{1+\cos\alpha}{2},$ and hence the formula for $\cos(\alpha/2)$. $\qquad \blacksquare$

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Topics: Trigonometric formulae
Concepts: Bisection formulae · Duplication formulae
Functions: Cosine · Sine
Methods: Bisection formulae
Skills: Proving