Sum and difference angle formulae

The cosine function is an example of an even function.

For an even function f(-x) = f(x).

The sine function is an example of an odd function.

For an odd function g(-x) = -g(x).

An even function times an even function is even

f(-x) f(-x) = f(x) f(x).

An odd function times an odd function is also even.

g(-x) g(-x) = g(x) g(x).

Only and odd function times an even function is odd.

f(-x) g(-x) = - f(x) g(x).

The sum and difference angle formulae are

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

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

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

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

These can be writted more succinctly as 

Note that one way to remember these formulae is the recall that cosine is even and for the product on the right hand side to remain even like functions must be multiplied. Similarly, sine is odd and therefore an odd function (sine) must multiply an even function (cosine).

The double angle formulae derive from the sum formulae.

If a = b, then

Finally, combinations of the sum and difference formulae can be used to generate expressions such as

Note that

Collecting terms and using the sin²a + cos²a = 1 identity we end up with the desired result.