Trigonometric Addition Formulas

Angle addition formulas express trigonometric functions of sums of angles in terms of functions of and . The fundamental formulas of angle addition in trigonometry are given by

The first four of these are known as the prosthaphaeresis formulas, or sometimes as Simpson's formulas.

The sine and cosine angle addition identities can be compactly summarized by the matrix equation

These formulas can be simply derived using complex exponentials and the Euler formula as follows.

Equating real and imaginary parts then gives (1) and (3), and (2) and (4) follow immediately by substituting for .

Taking the ratio of (1) and (3) gives the tangent angle addition formula

The double-angle formulas are

Multiple-angle formulas are given by

and can also be written using the recurrence relations

The angle addition formulas can also be derived purely algebraically without the use of complex numbers. Consider the small right triangle in the figure above, which gives

Now, the usual trigonometric definitions applied to the large right triangle give

Solving these two equations simultaneously for the variables and then immediately gives

These can be put into the familiar forms with the aid of the trigonometric identities

and

which can be verified by direct multiplication. Plugging (◇) into (◇) and (38) into (◇) then gives

as before.

A similar proof due to Smiley and Smiley uses the left figure above to obtain

from which it follows that

Similarly, from the right figure,

so

Similar diagrams can be used to prove the angle subtraction formulas (Smiley 1999, Smiley and Smiley). In the figure at left,

giving

Similarly, in the figure at right,

giving

A more complex diagram can be used to obtain a proof from the identity (Ren 1999). In the above figure, let . Then

An interesting identity relating the sum and difference tangent formulas is given by