Sum and Difference Formulas

The angle sum and difference formulas are useful because they allow certain angles to be expressed in trigonometric functions in two parts (x and y), which may make more complex calculations (such as integration) easier. The angle sum and difference formulas for sine and cosine are sometimes referred to as Simpson’s formulas. For the sum or difference of two angles x and y, the main trigonometric functions are:

sin (x + y) = (sin x · cos y) + (cos x · sin y)
sin (x – y) = (sin x · cos y) – (cos x · sin y)
cos (x + y) = (cos x · cos y) – (sin x · sin y)
cos (x – y) = (cos x · cos y) + (sin x · sin y)
tan (x + y) = tan x + tany / 1-tan x · tan y
tan (x – y) = tan x – tan y / 1+tan x · tan y

NOTE:

We can use the identity cot x = 1 / tan x to solve the problems involving cot x. Therefore we do not need to remember the sum and difference formulas for cot x.

Example: cos75° = ?

Solution: cos75° = cos (45° + 30°) = (cos45° · cos30°) – (sin45° · sin30°)
=√2/2 · √3/2 – √2/2 · 1/2 = √6 – √2 / 4