Sine and cosine don't distribute over addition — but they do obey their own precise rule for splitting a compound angle into two simpler ones.
9 min read4 worked examples
The addition formulae
It's tempting to think $\sin(A+B) = \sin A + \sin B$, but this is false in general. The correct identities, called the addition formulae (or compound angle formulae), are:
$\sin(A\pm B) = \sin A\cos B \pm \cos A\sin B$
$\cos(A\pm B) = \cos A\cos B \mp \sin A\sin B$
$\tan(A\pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$
The $\pm$ and $\mp$ symbols mean the formulae compress two identities into one line: read the top sign throughout for the "$+$" version, and the bottom sign throughout for the "$-$" version. Notice the sign flips in the cosine formula (a "$+$" on the left becomes a "$-$" on the right, and vice versa) — a common source of error.
Where these come from
These identities can be established geometrically, by constructing two right-angled triangles at angles $A$ and $B$ inside a unit circle (or a carefully constructed diagram of overlapping triangles) and comparing side lengths and areas. The full geometric argument is intricate, but the core idea is that rotating a unit vector first by angle $B$ then by angle $A$ lands it at angle $A+B$ — and expanding the coordinates of that rotated point using right-angled trigonometry on each stage produces exactly the $\cos A\cos B - \sin A\sin B$ and $\sin A\cos B+\cos A\sin B$ combinations seen above. At this level, the formulae are typically quoted and used directly rather than proved from scratch.
The tangent formula follows from the sine and cosine ones, since $\tan(A\pm B) = \dfrac{\sin(A\pm B)}{\cos(A\pm B)}$; dividing numerator and denominator through by $\cos A\cos B$ turns the sines and cosines into tangents.
Using the formulae for exact values
Angles that aren't among the standard exact-value angles ($30°,45°,60°,\dots$) can sometimes be built by adding or subtracting two that are — e.g. $75° = 45°+30°$, or $15° = 45°-30°$. Substituting into the addition formulae then produces an exact (often surd-heavy) value.
Worked example 1
Exact value via addition formula
Find the exact value of $\sin 75°$, giving your answer as a single simplified surd expression.
MethodWrite 75° as 45° + 30° and apply the sine addition formula
$\dfrac12\sin x = \cos x - \dfrac{\sqrt3}{2}\cos x = \left(1-\dfrac{\sqrt3}{2}\right)\cos x$.
Collect all cos x terms on one side.
4
Divide both sides by $\cos x$: $\dfrac12\tan x = 1-\dfrac{\sqrt3}{2} \Rightarrow \tan x = 2-\sqrt3$.
sin x / cos x = tan x; multiply both sides by 2.
5
$\tan x = 2-\sqrt3 \approx 0.268 \Rightarrow x = 15°$ (principal value); tan has period $180°$, so $x = 15°$ or $x=195°$.
Tangent repeats every 180°, giving two solutions in [0°, 360°).
▪ solved
Examiner's eye
Dividing by $\cos x$ is valid here since $\cos x=0$ would force $\sin x=0$ too (from the rearranged equation), which is impossible simultaneously — so no solutions are lost by dividing.
Worked example 4
Proving a further identity using the addition formulae · Challenge
Show that $\dfrac{\sin(A+B)}{\cos A\cos B} = \tan A + \tan B$.
MethodExpand the numerator, then split the fraction into two simpler ones
1
Start from the left-hand side: $\dfrac{\sin(A+B)}{\cos A\cos B} = \dfrac{\sin A\cos B + \cos A\sin B}{\cos A\cos B}$.
Expand the numerator using the sine addition formula.
2
Split into two fractions: $= \dfrac{\sin A\cos B}{\cos A\cos B} + \dfrac{\cos A\sin B}{\cos A\cos B}$.
A sum in the numerator over a shared denominator splits into a sum of fractions.
cos B cancels in the first fraction, cos A cancels in the second.
4
$= \tan A + \tan B$, which is the right-hand side.
sin/cos = tan by definition, for each angle.
▪ proven
Look deeper
This kind of derived identity is exactly the pattern used to build the tangent addition formula itself — dividing the sine and cosine addition formulae through by $\cos A\cos B$ and simplifying gives $\tan(A+B)=\dfrac{\tan A+\tan B}{1-\tan A\tan B}$ directly.
Common mistake
Writing $\sin(A+B) = \sin A + \sin B$ or $\cos(A+B)=\cos A+\cos B$ — trigonometric functions do not distribute over addition. Always expand a compound angle using the full addition formula, never by splitting the function across the sum.
Check yourself
Find the exact value of $\cos 15°$, giving your answer as a single simplified surd expression. (Hint: write 15° = 45° − 30° and use the cosine subtraction formula.)