# Sum-to-Product Identities

Sometimes we may need to simplify a trigonometric expression like sin α ± sin β or cos α ± cos β by converting the sum or difference of trigonometric functions into a product. These formulas are known as sum-to-product identities.

## Sum and Difference of Sines

To derive the sum-to-product identities for sine we use the sine addition and subtraction formulas:

$\sin \left( {x + y} \right) = \sin x \cos y + \cos x \sin y ,$
$\sin \left( {x - y} \right) = \sin x \cos y - \cos x \sin y .$

If we add and subtract the two equations, we get

$\sin \left( {x + y} \right) + \sin \left( {x - y} \right) = 2\sin x \cos y,$
$\sin \left( {x + y} \right) - \sin \left( {x - y} \right) = 2\cos x \sin y.$

Let now $$x = y = \alpha$$ and $$x - y = \beta.$$ Solving this system for $$x$$ and $$y,$$ we obtain

$x = \frac{{\alpha + \beta }}{2},\;\;y = \frac{{\alpha - \beta }}{2}.$

Plugging this into the equations above yields:

$\sin\alpha + \sin\beta = 2\sin\frac{\alpha + \beta}{2} \cos\frac{\alpha - \beta}{2}$
$\sin\alpha - \sin\beta = 2\cos\frac{\alpha + \beta}{2} \sin\frac{\alpha - \beta}{2}$

## Sum and Difference of Cosines

Similarly, using the cosine addition and subtraction formulas

$\cos \left( {x + y} \right) = \cos x\cos y - \sin x\sin y,$
$\cos \left( {x - y} \right) = \cos x\cos y + \sin x\sin y,$

we can derive the sum-to-product identities for cosine:

$\cos\alpha + \cos\beta = 2\cos\frac{\alpha + \beta}{2} \cos\frac{\alpha - \beta}{2}$
$\cos\alpha - \cos\beta = -2\sin\frac{\alpha + \beta}{2} \sin\frac{\alpha - \beta}{2}$

## Sum and Difference of Tangents

We have

$\tan \alpha + \tan \beta = \frac{{\sin \alpha }}{{\cos \alpha }} + \frac{{\sin \beta }}{{\cos \beta }} = \frac{{\sin \alpha \cos \beta + \cos \alpha \sin \beta }}{{\cos \alpha \cos \beta}}.$

The numerator in the last expression is equal to $$\sin \left( {\alpha + \beta } \right).$$ Hence,

$\tan\alpha + \tan\beta = \frac{\sin\left({\alpha + \beta}\right)}{\cos\alpha\cos\beta}$

This formula is valid if $$\cos\alpha \ne 0$$ and $$\cos\beta \ne 0,$$ or $$\alpha ,\beta \ne \frac{\pi }{2} + \pi n,$$ $$n \in \mathbb{Z}.$$

Consider the difference of tangents:

$\tan \alpha - \tan \beta = \frac{{\sin \alpha }}{{\cos \alpha }} - \frac{{\sin \beta }}{{\cos \beta }} = \frac{{\sin \alpha \cos \beta - \cos \alpha \sin \beta }}{{\cos \alpha \cos \beta }} = \frac{{\sin \left( {\alpha - \beta } \right)}}{{\cos \alpha \cos \beta }},$

that is,

$\tan\alpha - \tan\beta = \frac{\sin\left({\alpha - \beta}\right)}{\cos\alpha\cos\beta}$

## Sum and Difference of Cotangents

The sum-to-product formulas for cotangent can de derived in the same way. They are given by

$\cot\alpha + \cot\beta = \frac{\sin\left({\alpha + \beta}\right)}{\sin\alpha\sin\beta}$
$\cot\alpha - \cot\beta = -\frac{\sin\left({\alpha - \beta}\right)}{\sin\alpha\sin\beta}$

where $$\sin\alpha \ne 0$$ and $$\sin\beta \ne 0,$$ or $$\alpha ,\beta \ne \pi n,$$ $$n \in \mathbb{Z}.$$

See solved problems on Page 2.