Cosine of Sum

Theorem

$\map \cos {a + b} = \cos a \cos b - \sin a \sin b$

where $\sin$ denotes the sine and $\cos$ denotes the cosine.

Corollary

$\map \cos {a - b} = \cos a \cos b + \sin a \sin b$

Proof 1

 $\displaystyle \map \cos {a + b} + i \, \map \sin {a + b}$ $=$ $\displaystyle e^{i \paren {a + b} }$ Euler's Formula $\displaystyle$ $=$ $\displaystyle e^{i a} e^{i b}$ Exponential of Sum $\displaystyle$ $=$ $\displaystyle \paren {\cos a + i \sin a} \paren {\cos b + i \sin b}$ Euler's Formula $\displaystyle$ $=$ $\displaystyle \paren {\cos a \cos b - \sin a \sin b} + i \paren {\sin a \cos b + \cos a \sin b}$ Complex Numbers form Field

The result follows by equating the real parts.

$\blacksquare$

Proof 2

Recall the analytic definitions of sine and cosine:

$\displaystyle \sin x = \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2 n + 1}} {\left({2 n + 1}\right)!}$
$\displaystyle \cos x = \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2 n}} {\left({2 n}\right)!}$

Let:

 $\displaystyle g \left({a}\right)$ $=$ $\displaystyle \sin \left({a + b}\right) - \sin a \cos b - \cos a \sin b$ $\displaystyle h \left({a}\right)$ $=$ $\displaystyle \cos \left({a + b}\right) - \cos a \cos b + \sin a \sin b$

Let us differentiate these with respect to $a$, keeping $b$ constant.

Then from Derivative of Sine Function and Derivative of Cosine Function, we have:

 $\displaystyle g' \left({a}\right)$ $=$ $\displaystyle \cos \left({a + b}\right) - \cos a \cos b + \sin a \sin b = h \left({a}\right)$ $\displaystyle h' \left({a}\right)$ $=$ $\displaystyle - \sin \left({a + b}\right) + \sin a \cos b + \cos a \sin b = - g \left({a}\right)$

Hence:

 $\displaystyle D_a \left({\left({g \left({a}\right)}\right)^2 + \left({h \left({a}\right)}\right)^2}\right)$ $=$ $\displaystyle 2 g \left({a}\right) g' \left({a}\right) + 2 h \left({a}\right) h' \left({a}\right)$ $\displaystyle$ $=$ $\displaystyle 0$

Thus from Derivative of Constant:

$\forall a \in \R: g \left({a}\right)^2 + h \left({a}\right)^2 = c$

In particular, it is true for $a = 0$, and so:

$g \left({0}\right)^2 + h \left({0}\right)^2 = 0$

So:

$g \left({a}\right)^2 + h \left({a}\right)^2 = 0$
$g \left({a}\right)^2 \ge 0$

and:

$h \left({a}\right)^2 \ge 0$

So it follows that:

$g \left({a}\right) = 0$

and:

$h \left({a}\right) = 0$

Hence the result.

$\blacksquare$

Proof 3

 $\displaystyle \cos a \cos b - \sin a \sin b$ $=$ $\displaystyle \paren {\frac {e^{i a} + e^{-i a} } 2} \paren {\frac {e^{i b} + e^{-i b} } 2} - \sin a \sin b$ Cosine Exponential Formulation $\displaystyle$ $=$ $\displaystyle \paren {\frac {e^{i a} + e^{-i a} } 2} \paren {\frac {e^{i b} + e^{-i b} } 2} - \paren {\frac {e^{i a} - e^{-i a} } {2 i} } \paren {\frac {e^{i b} - e^{-i b} } {2 i} }$ Sine Exponential Formulation $\displaystyle$ $=$ $\displaystyle \frac {e^{i a} e^{i b} + e^{i a} e^{-i b} + e^{-i a} e^{i b} + e^{-i a} e^{-i b} } 4$ $\displaystyle$  $\, \displaystyle - \,$ $\displaystyle \frac {e^{i a} e^{i b} - e^{i a} e^{-i b} - e^{-i a} e^{i b} + e^{-i a} e^{-i b} } {4 i^2}$ $\displaystyle$ $=$ $\displaystyle \frac {e^{i a} e^{i b} + e^{i a} e^{-i b} + e^{-i a} e^{i b} + e^{-i a} e^{-i b} } 4$ $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle \frac {e^{i a} e^{i b} - e^{i a} e^{-i b} - e^{-i a} e^{i b} + e^{-i a} e^{-i b} } 4$ as $i^2 = -1$ $\displaystyle$ $=$ $\displaystyle \frac {e^{i a} e^{i b} + e^{-ia} e^{-ib} } 2$ $\displaystyle$ $=$ $\displaystyle \frac {e^{i \paren {a + b} } + e^{-i \paren {a + b} } } 2$ Exponential of Sum $\displaystyle$ $=$ $\displaystyle \map \cos {a + b}$ Cosine Exponential Formulation

$\blacksquare$

Proof 4

$AB$, $AC$, $AE$, and $AD$ are radii of the circle centered at $A$.

Let $\angle BAC = a$ and $\angle DAC = \angle BAE = b$.

By Euclid's First Postulate, we can construct line segments $BD$ and $CE$.

By Euclid's second common notion, $\angle DAB = \angle CAE$.

Thus by Triangle Side-Angle-Side Equality, $\triangle DAB \cong \triangle CAE$.

Therefore, $DB = CE$.

We now assign Cartesian coordinates to the points $B$, $C$, $D$, and $E$:

 $\displaystyle B$ $=$ $\displaystyle \tuple {1, 0}$ $\displaystyle C$ $=$ $\displaystyle \tuple {\cos a, \sin a}$ $\displaystyle D$ $=$ $\displaystyle \tuple {\map \cos {a + b}, \map \sin {a + b} }$ $\displaystyle E$ $=$ $\displaystyle \tuple {\cos b, -\sin b}$ Cosine Function is Even and Sine Function is Odd

We use the definition of the distance function on the Euclidean space $\struct {\R^2, d}$ as defined by the Euclidean metric:

$\forall x, y \in \R^2: \map d {x, y} = \sqrt {\paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2}$

where $x = \tuple {x_1, y_1}, y = \tuple {x_2, y_2}$.

Thus:

$DB \cong CE \iff \map d {D, B} = \map d {C, E}$

So, plugging in the coordinates of $B, C, D, E$, we get:

 $\displaystyle \paren {\map \cos {a + b} } - 1)^2 + \map {\sin^2} {a + b}$ $=$ $\displaystyle \paren {\cos a - \cos b}^2 + \paren {\sin a + \sin b}^2$ $\displaystyle \leadsto \ \$ $\displaystyle \cos^2 \left({a + b}\right) + \sin^2 \left({a + b}\right)$  $\displaystyle$ multiplying out left hand side $\displaystyle {} - 2 \, \map \cos {a + b} + 1$ $=$ $\displaystyle \paren {\cos a - \cos b}^2 + \paren {\sin a + \sin b}^2$ $\displaystyle \leadsto \ \$ $\displaystyle 1 - 2 \, \map \cos {a + b} + 1$ $=$ $\displaystyle \paren {\cos a - \cos b}^2 + \paren {\sin a + \sin b}^2$ Sum of Squares of Sine and Cosine $\displaystyle \leadsto \ \$ $\displaystyle 2 - 2 \, \map \cos {a + b}$ $=$ $\displaystyle \cos^2 a - 2 \cos a \cos b + \cos^2 b$ multiplying out right hand side $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle \sin^2 a + 2 \sin a \sin b + \sin^2 b$ $\displaystyle \leadsto \ \$ $\displaystyle 2 - 2 \, \map \cos {a + b}$ $=$ $\displaystyle 2 - 2 \cos a \cos b + 2 \sin a \sin b$ Sum of Squares of Sine and Cosine $\displaystyle \leadsto \ \$ $\displaystyle \map \cos {a + b}$ $=$ $\displaystyle \cos a \cos b - \sin a \sin b$

$\blacksquare$

Historical Note

The Cosine of Sum formula and its corollary were proved by François Viète in about $1579$.