# Sine and Cosine Sum of Angles From Euler's Formula

The proof of the formulas of the sine and cosine of the sum of angles, using Euler's formula , is presented.

## Euler's Formula

Euler's Formula is given by $e^{ix} = \cos x + i \sin x \qquad (1)$ where $x$ is a real number and $i = \sqrt{-1}$ is the imaginary unit.

## Sine and Cosine of Sums of Angles Formulas

Using Euler's formula with the argument $x + y$, we write
$\displaystyle e^{i(x+y)} = \cos (x + y) + i \sin (x + y) \qquad (2)$

Using properties of
exponential functions , we write
$\displaystyle e^{i(x+y)} = e^{ix} e^{iy} \qquad (3)$

The left sides of $(2)$ and $(3)$ are the same, hence we write
$\displaystyle \cos (x + y) + i \sin (x + y) = e^{ix} e^{iy} \qquad (4)$

Use Euler's formula to the terms on the right of $(4)$, we obtain
$\displaystyle \cos (x + y) + i \sin (x + y) = (\cos x + i \sin x) (\cos y + i \sin y) \qquad (5)$

Expand the right term of $(5)$ using the fact that $i^2 = - 1$, to obtain
$\displaystyle \cos (x + y) + i \sin (x + y) = \cos x \cos y - \sin x \sin y + i ( \sin x \cos y + \cos x \sin y ) \qquad (6)$
The expressions on the left and on the right of $(6)$ are equal if their real parts are equal and their imaginary parts are equal; hence the formula of sine and cosine of the sums of angles

$\cos (x + y) = \cos x \cos y - \sin x \sin y$ and $\sin (x + y) = \sin x \cos y + \cos x \sin y$