The **sum formulae** are given by

sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
cos(a-b) = cos(a)cos(b) + sin(a)sin(b).

By taking

linear combinations of these, we obtain the

**product formulae**.

2sin(a)cos(b) = sin(a+b) + sin(a-b)
2cos(a)cos(b) = cos(a+b) + cos(a-b)
2sin(a)sin(b) = cos(a-b) - cos(a+b).

These results can be proved using a geometric argument, or by using the

exponential forms of

cosine and

sine, namely

2cos(x) = exp(ix) + exp(-ix)
2isin(x) = exp(ix) - exp(-ix).

They are immensely useful across most of

applied mathematics. One of the most simple consequences of them are the

**double angle formulae**, obtained by putting b = a:

sin(2a) = 2sin(a)cos(a)
cos(2a) = cos^{2}(a) - sin^{2}(a).

They can also be used to verify de Moivre's Theorem.