Sum and Difference Identities

You know the unit circle gives sin and cos for every standard angle. But what about an angle that is not a standard one, like $75°$ or $\\pi/12$? If you can write the awkward angle as a sum or difference of two standard angles, the addition formulas convert that algebraic decomposition into an exact value. Beyond computation, they are the engine behind the double-angle formulas, the half-angle identities used in calculus integration, the rotation matrices in linear algebra, and the proof that sine and cosine satisfy the differential equation $y'' + y = 0$.

The four formulas

For any angles $A$ and $B$:

Notice the sign pattern: sine of a sum mixes sine and cosine with a plus, while cosine of a sum mixes sine-and-sine with a minus. The minus inside is the formula’s claim to fame, it is what makes $\\cos(A+B) \\neq \\cos A + \\cos B$.

Worked example: cos 15°

$15° = 45° - 30°$, both standard. Apply the cosine difference formula:

Exact, no calculator. The same trick handles $\\sin 75°$, $\\tan 75°$, and any angle expressible as a sum or difference of $30°$, $45°$, $60°$, $90°$.

The double-angle formulas

Set $A = B = \\theta$ in the sum formulas:

The cosine version has three equivalent forms thanks to the Pythagorean identity. Each is the right one in a different context. Rearranged, they become the half-angle formulas $\\cos^2\\theta = (1 + \\cos 2\\theta)/2$ and $\\sin^2\\theta = (1 - \\cos 2\\theta)/2$, indispensable when integrating $\\sin^2 x$ or $\\cos^2 x$ in calculus.

Tangent of a sum

Dividing the sine sum by the cosine sum and simplifying yields:

valid whenever the denominator is nonzero. The denominator vanishes precisely when $A + B = \\pi/2 + n\\pi$, right where $\\tan(A+B)$ has an asymptote, as it should.

• Signal Processing: Modulating a carrier signal with audio uses $\\sin(\\omega\_c t + m(t))$; expanding via the addition formula gives the sum-of-frequencies form that AM/FM radio engineers use to design tuner circuits.
• Computer Graphics: Rotating a model by angle $\\alpha + \\beta$ instead of by $\\alpha$ then by $\\beta$ should give the same result; the addition formulas are the algebraic statement of that geometric consistency.
• Quantum Mechanics: Spin-half rotation matrices contain $\\cos(\\theta/2)$ and $\\sin(\\theta/2)$; the double-angle formula is what lets physicists derive that a spin-1/2 particle returns to itself only after a full 720^\\circ rotation.

(Use the unit-circle widget to verify your worked answers. Snap to $\\pi/4$ to read off $\\sin(\\pi/4)$ and $\\cos(\\pi/4)$; the addition formulas combine these with values at $\\pi/6$ to give exact answers at $5\\pi/12$.)

Try it

• Compute $\\sin 75°$ using $75° = 45° + 30°$.
• Given $\\sin\\theta = 3/5$ and $\\theta \\in$ QI, find $\\sin(2\\theta)$ and $\\cos(2\\theta)$. (First find $\\cos\\theta = 4/5$ from the Pythagorean identity, then plug in.)
• Show that $\\sin(A+B) - \\sin(A-B) = 2\\cos A \\sin B$. (Subtract the two sine formulas; the $\\sin A\\cos B$ terms cancel.)

Pause: in the cosine formula, why is the sign rule reversed compared to sine? Hint: the cosine wave has different symmetry, it is even ($\\cos(-\\theta) = \\cos\\theta$) while sine is odd ($\\sin(-\\theta) = -\\sin\\theta$). Substitute $-B$ for $B$ in $\\cos(A + B)$ and see what happens.

A trap to watch for

The sign in the cosine sum formula reverses for a difference, but in the opposite direction from sine:

$\\cos(A - B) = \\cos A\\cos B + \\sin A\\sin B$, the second term is plus, not minus.

Beginners routinely write $\\cos(A-B) = \\cos A\\cos B - \\sin A\\sin B$, copying the sum formula. Use this memory device: for cosine, sum gets a minus, difference gets a plus; for sine, sum gets a plus, difference gets a minus. The reason is the odd/even symmetry of sine and cosine: substituting $-B$ for $B$ flips the sine but not the cosine, which is exactly why the rule flips.

What you now know

You can state all four addition formulas, derive the double-angle formulas as the special case $A = B$, compute exact values for angles like $75°$ or $15°$ that are not on the standard table, and avoid the cosine sign-flip trap. The next section uses these formulas as the algebraic backbone of rotations, turning the plane around the origin via a $2 \\times 2$ matrix whose entries are sin and cos.

Quick check

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References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 11, Section 5, derivation of the addition formulas from the unit-circle definition.
• Spivak, M. (2008). Calculus (4th ed.). Publish or Perish. Chapter 15: addition formulas derived from the differential equation $y'' = -y$.
• Apostol, T. M. (1967). Calculus, Volume 1 (2nd ed.). Wiley. Section 2.5, algebraic and geometric proofs of the sum-of-angles identities.