The proof of the formulas of the sine and cosine of the sum of angles, using Euler's formula , is presented.
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.
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
\[ \sin (x + y) = \sin x \cos y + \cos x \sin y \]
