Angle Addition &
Subtraction Formulas
Master the proofs and applications of the sine, cosine, and tangent addition and subtraction identities: the building blocks of trigonometric reasoning.
Trigonometry
What Are Angle Sum Formulas?
The angle addition and subtraction formulas express the sine, cosine, or tangent of a sum or difference of two angles in terms of the trig functions of each angle individually.
These identities let us find exact values of trig functions at non-standard angles (like \(75^\circ\) or \(15^\circ\)) and form the foundation for double-angle, half-angle, and product-to-sum formulas.
Why These Identities Matter
Exact Values
Calculate \(\cos 15^\circ\) or \(\sin 75^\circ\) without a calculator by decomposing into known angles.
Proof Engine
Derive every other trig identity: double-angle, half-angle, product-to-sum, and sum-to-product.
Real-World
Used in physics (wave interference), engineering (signal processing), and computer graphics (rotations).
Visual Setup: The Unit Circle
The Core Formulas
$$\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$$
$$\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$$
$$\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}$$
Proof of \(\cos(\alpha - \beta)\)
1
Place two points on the unit circle: \(A = (\cos\alpha, \sin\alpha)\) at angle \(\alpha\) and \(B = (\cos\beta, \sin\beta)\) at angle \(\beta\).
2
Compute \(|AB|^2\) via distance formula: \((\cos\alpha - \cos\beta)^2 + (\sin\alpha - \sin\beta)^2\). Expand to get \(2 - 2(\cos\alpha\cos\beta + \sin\alpha\sin\beta)\).
3
Rotate the picture so that \(B\) lands on \((1,0)\). Then \(A\) moves to \((\cos(\alpha-\beta), \sin(\alpha-\beta))\). The squared distance is now \(2 - 2\cos(\alpha-\beta)\).
4
Set the two expressions equal: since rotation preserves distance, \(\cos(\alpha-\beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta\). ∎
The Distance Proof: Visualized
Deriving All Other Formulas
1
\(\cos(\alpha + \beta)\): Replace \(\beta\) with \(-\beta\) in the subtraction formula. Since \(\cos(-\beta)=\cos\beta\) and \(\sin(-\beta)=-\sin\beta\), we get \(\cos\alpha\cos\beta - \sin\alpha\sin\beta\).
2
\(\sin(\alpha + \beta)\): Use the cofunction identity \(\sin\theta = \cos\!\bigl(\frac{\pi}{2}-\theta\bigr)\). Write \(\sin(\alpha+\beta) = \cos\!\bigl(\frac{\pi}{2}-\alpha-\beta\bigr)\), then apply the cosine subtraction formula.
3
\(\sin(\alpha - \beta)\): Replace \(\beta\) with \(-\beta\) in the sine addition formula.
4
\(\tan(\alpha \pm \beta)\): Divide \(\sin(\alpha\pm\beta)\) by \(\cos(\alpha\pm\beta)\), then divide numerator and denominator by \(\cos\alpha\cos\beta\).
Addition vs Subtraction
| Function | Addition \((\alpha+\beta)\) | Subtraction \((\alpha-\beta)\) |
|---|---|---|
| Cosine | \(\cos\alpha\cos\beta\) \(-\) \(\sin\alpha\sin\beta\) | \(\cos\alpha\cos\beta\) \(+\) \(\sin\alpha\sin\beta\) |
| Sine | \(\sin\alpha\cos\beta\) \(+\) \(\cos\alpha\sin\beta\) | \(\sin\alpha\cos\beta\) \(-\) \(\cos\alpha\sin\beta\) |
| Tangent | \(\dfrac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}\) | \(\dfrac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta}\) |
How to Apply the Formulas
1
Decompose the angle into a sum or difference of "nice" angles whose exact trig values you know: \(30^\circ, 45^\circ, 60^\circ, 90^\circ\) (or \(\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\)).
2
Choose the right formula: sine addition, cosine subtraction, etc. Watch the sign pattern carefully.
3
Substitute the known values for \(\sin\) and \(\cos\) of each component angle.
4
Simplify the arithmetic. Rationalize denominators if needed.
Example 1: \(\cos 75^\circ\)
Find the exact value of \(\cos 75^\circ\).
1
Decompose: \(75^\circ = 45^\circ + 30^\circ\).
2
Apply cosine addition: \(\cos(45^\circ+30^\circ) = \cos 45^\circ\cos 30^\circ - \sin 45^\circ\sin 30^\circ\).
3
Substitute: \(= \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2}\cdot\frac{1}{2}\).
4
Simplify: \(= \frac{\sqrt{6}-\sqrt{2}}{4}\).
Example 2: \(\sin 15^\circ\)
Find the exact value of \(\sin 15^\circ\).
1
Decompose: \(15^\circ = 45^\circ - 30^\circ\).
2
Apply sine subtraction: \(\sin(45^\circ - 30^\circ) = \sin 45^\circ\cos 30^\circ - \cos 45^\circ\sin 30^\circ\).
3
Substitute: \(= \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2}\cdot\frac{1}{2}\).
4
Simplify: \(= \frac{\sqrt{6}-\sqrt{2}}{4}\).
Example 3: \(\tan 105^\circ\)
Find the exact value of \(\tan 105^\circ\).
1
Decompose: \(105^\circ = 60^\circ + 45^\circ\).
2
Apply tangent addition: \(\tan(60^\circ+45^\circ) = \frac{\tan 60^\circ + \tan 45^\circ}{1 - \tan 60^\circ\tan 45^\circ}\).
3
Substitute: \(= \frac{\sqrt{3}+1}{1-\sqrt{3}}\).
4
Rationalize: Multiply by \(\frac{1+\sqrt{3}}{1+\sqrt{3}}\) to get \(\frac{(\sqrt{3}+1)(1+\sqrt{3})}{(1-\sqrt{3})(1+\sqrt{3})} = \frac{4+2\sqrt{3}}{-2} = -2-\sqrt{3}\).
Example 4: Proving an Identity
Prove: \(\sin\!\left(\frac{\pi}{2} - x\right) = \cos x\).
1
Apply sine subtraction with \(\alpha = \frac{\pi}{2}\) and \(\beta = x\):
2
\(\sin\!\bigl(\frac{\pi}{2} - x\bigr) = \sin\frac{\pi}{2}\cos x - \cos\frac{\pi}{2}\sin x\).
3
Evaluate known values: \(\sin\frac{\pi}{2}=1\) and \(\cos\frac{\pi}{2}=0\).
4
Simplify: \(= (1)\cos x - (0)\sin x = \cos x\). ∎
Common Mistakes to Avoid
✕ Distributing Trig
Writing \(\sin(\alpha+\beta) = \sin\alpha + \sin\beta\). Trig functions are not linear: you cannot distribute them over addition.
✕ Wrong Sign Pattern
Using \(+\) in the cosine addition formula instead of \(-\). Remember: cosine is the "contrarian" that flips the sign.
✕ Mixing Up Formulas
Using the sine structure (\(\sin\cos \pm \cos\sin\)) when computing cosine (which needs \(\cos\cos \mp \sin\sin\)).
Quick Reference Summary
| Formula | Sign Rule | Structure |
|---|---|---|
| \(\cos(\alpha\pm\beta)\) | Flips (\(\mp\)) | \(\cos\alpha\cos\beta \mp \sin\alpha\sin\beta\) |
| \(\sin(\alpha\pm\beta)\) | Keeps (\(\pm\)) | \(\sin\alpha\cos\beta \pm \cos\alpha\sin\beta\) |
| \(\tan(\alpha\pm\beta)\) | Numerator keeps, denom flips | \(\frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}\) |
| Proof method | Distance on unit circle | Rotation preserves distance |
Question 1 of 5
Which expression is equal to \(\cos(\alpha + \beta)\)?
The cosine addition formula flips the sign: \(\cos(\alpha+\beta)=\cos\alpha\cos\beta - \sin\alpha\sin\beta\). Choice (a) is the subtraction formula, and choices (c)/(d) are the sine formulas.
Question 2 of 5
What is the exact value of \(\sin 75^\circ\)?
Write \(75^\circ = 45^\circ+30^\circ\). Then \(\sin 75^\circ = \sin 45^\circ\cos 30^\circ+\cos 45^\circ\sin 30^\circ = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\cdot\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\). Choice (a) is \(\cos 75^\circ\).
Question 3 of 5
In the unit-circle proof of \(\cos(\alpha-\beta)\), what geometric fact guarantees the two distance expressions are equal?
The proof rotates the entire configuration by \(-\beta\). A rotation is an isometry (distance-preserving transformation), so the chord length is unchanged. That is why the two squared-distance expressions can be set equal.
Question 4 of 5
What is the exact value of \(\tan(45^\circ - 30^\circ) = \tan 15^\circ\)?
Using the tangent subtraction formula: \(\tan 15^\circ = \frac{\tan 45^\circ-\tan 30^\circ}{1+\tan 45^\circ\tan 30^\circ}=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}=\frac{\sqrt{3}-1}{\sqrt{3}+1}\). Rationalizing gives \(\frac{(\sqrt{3}-1)^2}{2}=\frac{4-2\sqrt{3}}{2}=2-\sqrt{3}\).
Question 5 of 5
Which identity is proven by setting \(\alpha = \beta\) in the cosine addition formula?
Setting \(\beta=\alpha\) in \(\cos(\alpha+\beta)=\cos\alpha\cos\beta - \sin\alpha\sin\beta\) gives \(\cos 2\alpha = \cos^2\alpha - \sin^2\alpha\). Choice (a) comes from the sine addition formula, and (c) is an algebraic rearrangement of (b), not the direct result of substitution.