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Toolkit 39

Trigonometric Sum-to-Product Identities

sin⁡α+sin⁡β=2sin⁡(α+β2)cos⁡(α−β2)\sin\alpha+\sin\beta=2\sin\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)
sin⁡α−sin⁡β=2cos⁡(α+β2)sin⁡(α−β2)\sin\alpha-\sin\beta=2\cos\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)
cos⁡α+cos⁡β=2cos⁡(α+β2)cos⁡(α−β2)\cos\alpha+\cos\beta=2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)
cos⁡α−cos⁡β=−2sin⁡(α+β2)sin⁡(α−β2)\cos\alpha-\cos\beta=-2\sin\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)

Proof

From Toolkit 38,

sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin(x+y) = \sin x\cos y + \cos x\sin y

and

sin⁡(x−y)=sin⁡xcos⁡y−cos⁡xsin⁡y.\sin(x-y) = \sin x\cos y - \cos x\sin y.

Adding these equations gives

sin⁡(x+y)+sin⁡(x−y)=2sin⁡xcos⁡y.\sin(x+y) + \sin(x-y) = 2\sin x\cos y.

Let

x=α+β2,y=α−β2.x = \frac{\alpha+\beta}{2}, \qquad y = \frac{\alpha-\beta}{2}.

Then x+y=αx+y = \alpha and x−y=βx-y = \beta. Therefore,

sin⁡α+sin⁡β=2sin⁡(α+β2)cos⁡(α−β2).\sin\alpha + \sin\beta = 2\sin\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right).

Replacing β\beta by −β-\beta, and using Toolkit 37, gives

sin⁡α−sin⁡β=2cos⁡(α+β2)sin⁡(α−β2).\sin\alpha - \sin\beta = 2\cos\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right).

Also, from Toolkit 38,

cos⁡(x+y)=cos⁡xcos⁡y−sin⁡xsin⁡y\cos(x+y) = \cos x\cos y - \sin x\sin y

and

cos⁡(x−y)=cos⁡xcos⁡y+sin⁡xsin⁡y.\cos(x-y) = \cos x\cos y + \sin x\sin y.

Adding gives

cos⁡(x−y)+cos⁡(x+y)=2cos⁡xcos⁡y.\cos(x-y) + \cos(x+y) = 2\cos x\cos y.

Using the same substitutions,

cos⁡α+cos⁡β=2cos⁡(α+β2)cos⁡(α−β2).\cos\alpha + \cos\beta = 2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right).

Subtracting the two cosine equations gives

cos⁡(x+y)−cos⁡(x−y)=−2sin⁡xsin⁡y.\cos(x+y) - \cos(x-y) = -2\sin x\sin y.

Therefore,

cos⁡α−cos⁡β=−2sin⁡(α+β2)sin⁡(α−β2).□\cos\alpha - \cos\beta = -2\sin\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right). \quad\square