Trigonometry/Multiple-Angle and Product-to-sum Formulas
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[edit] Proofs for Double Angle Formulas
[edit] Using the Sum and Difference Identities
Recall that:
Using a = b in the above formula yields:
From the last (rightmost) term, two more identities may be derived. One containing only a sine:
![\displaystyle\cos(2a) = \cos^2(a) - \sin^2(a) = \cos^2(a) - \sin^2(a) + 0 = \cos^2(a) - \sin^2(a) + [\sin^2(a) - \sin^2(a)] = [\cos^2(a) + \sin^2(a)] - 2\sin^2(a) = 1 - 2\sin^2(a)](http://upload.wikimedia.org/math/1/4/6/1465305461ecd171217ca8c16d4d3a83.png)
And one containing only a cosine:
![\displaystyle\cos(2a) = \cos^2(a) - \sin^2(a) = \cos^2(a) - \sin^2(a) + 0 = \cos^2(a) - \sin^2(a) + [\cos^2(a) - \cos^2(a)] = 2\cos^2(a) - [\cos^2(a) + \sin^2(a)] = 2\cos^2(a) - 1](http://upload.wikimedia.org/math/8/5/a/85a73f6ae4c9df62303d08e4729ed2df.png)
; a = b for sin(2a)
