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Half-Angle and Double Angle Formulas

Using the addition formulas we've learned, we can develop new identities for sin and cos that can be used to express the sine and cosine of half-angles in terms of the cosine of a whole angle. Recall the addition formula:

(1) - cos (a + b) = (cos a)(cos b) – (sin a)(sin b)

Replace a and b by x/2.

cos(x/2 + x/2) = (cos x/2) - (sin x/2)(sin x/2)


cos(x) = (cos x/2) squared = sin (x/2) squared

Using the identity (sin a)2 + (cos a)2 = 1, replace 1 - (sin x/2) squared by (cos x/2) squared in (2):