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:
If we add and subtract the two equations, we get
Let now \(x = y = \alpha\) and \(x - y = \beta.\) Solving this system for \(x\) and \(y,\) we obtain
Plugging this into the equations above yields:
Sum and Difference of Cosines
Similarly, using the cosine addition and subtraction formulas
we can derive the sum-to-product identities for cosine:
Sum and Difference of Tangents
We have
The numerator in the last expression is equal to \(\sin \left( {\alpha + \beta } \right).\) Hence,
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:
that is,
Sum and Difference of Cotangents
The sum-to-product formulas for cotangent can de derived in the same way. They are given by
where \(\sin\alpha \ne 0\) and \(\sin\beta \ne 0,\) or \(\alpha ,\beta \ne \pi n,\) \(n \in \mathbb{Z}.\)