The compound angle formula can be proved using simple trigonometry. Consider the right angled triangle below with internal right angled triangles constructed with internal anglesand

With sides labelledandas shown below,

we can find sides in terms of x and h.

Thenand

By labelling extra angles as below,

we deduce that

Equating these expressions for x gives

Substituteto give

h now cancels from both sides to give