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.
