Trigonometric identities (addition formulas)

Intuition [Beginner]

You know the sine and cosine of special angles like $30^{\circ }$, $45^{\circ }$, and $60^{\circ }$. The addition formulas let you compute the sine and cosine of sums like $75^{\circ }=30^{\circ }+45^{\circ }$ using only the values you already know.

The formulas are:

• $\sin (x+y)=\sin x\cos y+\cos x\sin y$
• $\cos (x+y)=\cos x\cos y-\sin x\sin y$

These two identities generate every other trigonometric identity. Double-angle formulas, half-angle formulas, sum-to-product formulas — they are all consequences of the addition formulas. The reason they work is geometric: rotating by angle $x$ and then by angle $y$ is the same as rotating by angle $x+y$, and the addition formulas record how the coordinates change under this combined rotation.

Visual [Beginner]

Two unit circles, one showing a rotation by angle $x$ from the point $(1,0)$ to $P=(\cos x,\sin x)$, and another showing a subsequent rotation by angle $y$ from $P$ to $Q=(\cos (x+y),\sin (x+y))$. The combined rotation takes $(1,0)$ to $Q$ in one step.

The addition formula for cosine tells you the $x$-coordinate of $Q$ in terms of the coordinates of the intermediate point $P$.

Worked example [Beginner]

Compute $\sin 75^{\circ }$ exactly.

Step 1. Write $75^{\circ }=30^{\circ }+45^{\circ }$.

Step 2. Apply the addition formula: $\sin (30^{\circ }+45^{\circ })=\sin 30^{\circ }\cos 45^{\circ }+\cos 30^{\circ }\sin 45^{\circ }$.

Step 3. Substitute known values: $=(1/2)(\sqrt{2}/2)+(\sqrt{3}/2)(\sqrt{2}/2)=\sqrt{2}/4+\sqrt{6}/4=(\sqrt{2}+\sqrt{6})/4$.

What this tells us: the addition formula produces exact values for angles that are sums of known angles, with no calculator needed.

Check your understanding [Beginner]

Formal definition [Intermediate+]

The addition formulas (also called sum-to-function formulas) are:

Definition. For all real numbers $x$ and $y$:

$$\sin (x+y)=\sin x\cos y+\cos x\sin y$$

$$\cos (x+y)=\cos x\cos y-\sin x\sin y$$

$$\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x\tan y}(\text{when }1-\tan x\tan y\ne 0)$$

The tangent addition formula follows from dividing the sine formula by the cosine formula.

Standard consequences. Setting $y=x$ gives the double-angle formulas:

$$\sin 2x=2\sin x\cos x,\cos 2x={\cos }^{2}x-{\sin }^{2}x=2{\cos }^{2}x-1=1-2{\sin }^{2}x.$$

Solving the last two forms for ${\cos }^{2}x$ and ${\sin }^{2}x$ gives the half-angle formulas:

$${\cos }^{2}x=\frac{1+\cos 2x}{2},{\sin }^{2}x=\frac{1-\cos 2x}{2}.$$

Counterexamples to common slips

• $\cos (x+y)\ne \cos x+\cos y$. The cosine of a sum is not the sum of the cosines. For $x=y=\pi /2$: $\cos (\pi /2+\pi /2)=\cos \pi =-1$, but $\cos (\pi /2)+\cos (\pi /2)=0+0=0$.
• $\sin (2x)\ne 2\sin x$ in general. For $x=\pi /3$: $\sin (2\pi /3)=\sqrt{3}/2\ne 2\cdot \sqrt{3}/2=\sqrt{3}$.
• The sign in $\cos (x+y)$ is minus. The sign in $\sin (x+y)$ is plus. The minus sign in the cosine formula is the source of all sign alternations in the derived identities.

Key theorem with proof [Intermediate+]

Theorem (addition formulas). For all real $x$ and $y$:

$$\cos (x+y)=\cos x\cos y-\sin x\sin y,$$

$$\sin (x+y)=\sin x\cos y+\cos x\sin y.$$

Proof. The rotation of the plane by angle $\alpha$ counter-clockwise is represented by the matrix:

$$R(\alpha )=\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right).$$

Rotating by $x$ and then by $y$ is the same as rotating by $x+y$, so $R(y)\cdot R(x)=R(x+y)$. Compute the matrix product on the left:

$$R(y)\cdot R(x)=\left(\begin{array}{cc}\cos y& -\sin y\\ \sin y& \cos y\end{array}\right)\left(\begin{array}{cc}\cos x& -\sin x\\ \sin x& \cos x\end{array}\right).$$

The $(1,1)$ entry is $\cos y\cos x-\sin y\sin x$. The $(1,1)$ entry of $R(x+y)$ is $\cos (x+y)$. Equating:

$$\cos (x+y)=\cos x\cos y-\sin x\sin y.$$

The $(2,1)$ entry is $\sin y\cos x+\cos y\sin x$. The $(2,1)$ entry of $R(x+y)$ is $\sin (x+y)$. Equating:

$$\sin (x+y)=\sin x\cos y+\cos x\sin y.\square$$

Bridge. The foundational reason the addition formulas hold is that rotation matrices compose by matrix multiplication, and the central insight is that the entries of the rotation matrix are $\cos \alpha$ and $\sin \alpha$. This is exactly the content that identifies trigonometric addition with the group law of rotations, and the bridge is between the analytic identities (formulas involving $\sin$ and $\cos$) and the geometric composition of rotations (matrix multiplication). Putting these together, every trigonometric identity is a consequence of $R(\alpha )R(\beta )=R(\alpha +\beta )$, and this pattern generalises to the rotation group $\mathrm{SO}(2)$ and its representation theory. The addition formulas build toward 00.05.03 where the Euler identity $e^{i(x+y)}=e^{ix}{e}^{iy}$ encodes both addition formulas simultaneously, and appears again in 00.07.01 where the unit-circle definition makes the rotation interpretation immediate.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (all four sum-to-product identities). For all real $A$, $B$:

$$\sin A+\sin B=2\sin \frac{A+B}{2}\cos \frac{A-B}{2}$$

$$\sin A-\sin B=2\cos \frac{A+B}{2}\sin \frac{A-B}{2}$$

$$\cos A+\cos B=2\cos \frac{A+B}{2}\cos \frac{A-B}{2}$$

$$\cos A-\cos B=-2\sin \frac{A+B}{2}\sin \frac{A-B}{2}$$

These are obtained by adding and subtracting the addition formulas evaluated at $(A+B)/2+(A-B)/2$ and $(A+B)/2-(A-B)/2$.

Theorem 2 (all four product-to-sum identities). For all real $A$, $B$:

$$\sin A\sin B=\frac{1}{2}[\cos (A-B)-\cos (A+B)]$$

$$\cos A\cos B=\frac{1}{2}[\cos (A-B)+\cos (A+B)]$$

$$\sin A\cos B=\frac{1}{2}[\sin (A+B)+\sin (A-B)]$$

$$\cos A\sin B=\frac{1}{2}[\sin (A+B)-\sin (A-B)]$$

These are the algebraic inverses of the sum-to-product formulas and are used to integrate products of trigonometric functions.

Theorem 3 (Chebyshev polynomials). The Chebyshev polynomial $T_n(x)$ of the first kind is defined by $T_n(\cos \theta )=\cos (n\theta )$. The first few are $T_0=1$, $T_1=x$, $T_2=2x^2-1$, and they satisfy the recurrence $T_{n+1}(x)=2x{T}_n(x)-T_{n-1}(x)$. This recurrence is the addition formula $\cos ((n+1)\theta )=2\cos \theta \cos (n\theta )-\cos ((n-1)\theta )$ rewritten in terms of $x=\cos \theta$.

Theorem 4 (multiple-angle formulas). The triple-angle formulas are:

$$\sin 3x=3\sin x-4{\sin }^{3}x,\cos 3x=4{\cos }^{3}x-3\cos x.$$

These are the case $n=3$ of the Chebyshev identities $\sin (nx)=U_{n-1}(\cos x)\sin x$ and $\cos (nx)=T_n(\cos x)$.

Theorem 5 (prosthaphaeresis). Before the invention of logarithms, the product-to-sum formulas were used to convert multiplication into addition for astronomical computation. The identity $\cos A\cos B=\frac{1}{2}[\cos (A-B)+\cos (A+B)]$ converts a product of two cosines into a sum of two cosines. Given a cosine table, one can compute products by table lookup and addition, avoiding multiplication entirely. This technique, called prosthaphaeresis (Greek for "addition and subtraction"), was used by astronomers including Tycho Brahe and was a direct precursor to Napier's logarithms (1614).

Theorem 6 (addition formulas from Euler's identity). The addition formulas are immediate consequences of $e^{i(x+y)}=e^{ix}{e}^{iy}$. Expanding both sides: $\cos (x+y)+i\sin (x+y)=(\cos x+i\sin x)(\cos y+i\sin y)=(\cos x\cos y-\sin x\sin y)+i(\sin x\cos y+\cos x\sin y)$. Equating real and imaginary parts gives both addition formulas simultaneously.

Synthesis. The foundational reason the addition formulas generate all trigonometric identities is that every trigonometric identity is a consequence of the group law $e^{ix}{e}^{iy}=e^{i(x+y)}$ for the circle group. The central insight is that the rotation matrices $R(\theta )$ form the group $\mathrm{SO}(2)$, and the addition formulas are the matrix entries of the product $R(x)R(y)=R(x+y)$. This is exactly the content that identifies the trigonometric identity system with the representation theory of the circle group, and the bridge is between the elementary formulas of precalculus and the representation theory of compact Lie groups. Putting these together, the double-angle and half-angle formulas are the specialisation $y=x$ and the identity $\cos (nx)=T_n(\cos x)$ is the $n$-th power of the rotation matrix, and this pattern recurs throughout the Chebyshev theory, the prosthaphaeresis tradition, and the Fourier decomposition. The addition formulas build toward 00.05.03 where the single identity $e^{i(x+y)}=e^{ix}{e}^{iy}$ unifies all of trigonometry, and the Chebyshev polynomials appear again in 00.07.01 where the unit-circle framework identifies $\cos (nx)$ with the Chebyshev evaluation $T_n(\cos x)$.

Full proof set [Master]

Proposition 1 (cosine addition formula). $\cos (x+y)=\cos x\cos y-\sin x\sin y$ for all real $x$, $y$.

Proof. The rotation matrix $R(\theta )=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$ satisfies $R(\alpha )R(\beta )=R(\alpha +\beta )$ because composing rotations by $\alpha$ and $\beta$ is a rotation by $\alpha +\beta$. Computing the matrix product: the $(1,1)$ entry of $R(y)R(x)$ is $\cos y\cos x-\sin y\sin x$, and the $(1,1)$ entry of $R(x+y)$ is $\cos (x+y)$. Equating gives the result. $\square$

Proposition 2 (sine subtraction formula). $\sin (x-y)=\sin x\cos y-\cos x\sin y$ for all real $x$, $y$.

Proof. Replace $y$ with $-y$ in $\sin (x+y)=\sin x\cos y+\cos x\sin y$. Since $\cos (-y)=\cos y$ and $\sin (-y)=-\sin y$: $\sin (x-y)=\sin x\cos (-y)+\cos x\sin (-y)=\sin x\cos y-\cos x\sin y$. $\square$

Proposition 3 (Chebyshev recurrence). If $T_n(\cos \theta )=\cos (n\theta )$, then $T_{n+1}(x)=2x{T}_n(x)-T_{n-1}(x)$.

Proof. From the cosine addition formula: $\cos ((n+1)\theta )=\cos (n\theta +\theta )=\cos (n\theta )\cos \theta -\sin (n\theta )\sin \theta$. Also, $\cos ((n-1)\theta )=\cos (n\theta -\theta )=\cos (n\theta )\cos \theta +\sin (n\theta )\sin \theta$. Adding: $\cos ((n+1)\theta )+\cos ((n-1)\theta )=2\cos (n\theta )\cos \theta$. Setting $x=\cos \theta$: $T_{n+1}(x)+T_{n-1}(x)=2x{T}_n(x)$, hence $T_{n+1}(x)=2x{T}_n(x)-T_{n-1}(x)$. $\square$

Connections [Master]

• Unit-circle trigonometry 00.07.01. The addition formulas are the algebraic expression of the geometric fact that the composition of two unit-circle rotations $R(x)$ and $R(y)$ is the rotation $R(x+y)$. The unit-circle definition of $\sin$ and $\cos$ as coordinates from 00.07.01 makes this interpretation immediate: the addition formulas are the matrix entries of the product of two rotation matrices.

• Complex numbers and Euler's identity 00.05.03. The single identity $e^{i(x+y)}=e^{ix}\cdot e^{iy}$ generates both the sine and cosine addition formulas simultaneously when the real and imaginary parts are equated. This is the foundational reason the addition formulas work: they are the real and imaginary components of the exponential law, and every trigonometric identity is a shadow of the complex-exponential identity.

• Right-triangle trigonometry 00.06.01. The counterexample $\sin (2x)\ne 2\sin x$ in 00.06.01 is resolved by the addition formulas: $\sin (2x)=2\sin x\cos x$, which reduces to $2\sin x$ only when $\cos x=1$ (i.e., $x=0$). The correct nonlinear relationship between $\sin (2x)$ and $\sin x$ is the first indication that trigonometric functions are not linear, and the addition formulas provide the exact nonlinear correction.

Historical & philosophical context [Master]

Ptolemy ~150 CE Almagest ^[Ptolemy150] proved the chord addition formula $\mathrm{crd}(\alpha +\beta )$ in terms of $\mathrm{crd}(\alpha )$, $\mathrm{crd}(\beta )$, and $\mathrm{crd}(\alpha -\beta )$, which is equivalent to the sine subtraction formula when $\mathrm{crd}(\theta )=2\sin (\theta /2)$. The modern algebraic form of the addition formulas is due to Euler 1748 Introductio in analysin infinitorum ^[Euler1748], who derived them from the product formula for the complex exponential. The product-to-sum identities (prosthaphaeresis) were developed by Werner 1514 and used extensively by Brahe for astronomical calculations in the late 16th century, preceding Napier's logarithms as the primary computational tool for reducing multiplication to addition. The Chebyshev polynomials were introduced by Chebyshev 1854 in the context of approximation theory; the identity $T_n(\cos \theta )=\cos (n\theta )$ connects them directly to the addition formulas.

Bibliography [Master]

@book{Ptolemy150,
  author = {Ptolemy, Claudius},
  title = {Mathematike Syntaxis (Almagest)},
  year = {~150 CE}
}

@book{Euler1748,
  author = {Euler, Leonhard},
  title = {Introductio in analysin infinitorum},
  publisher = {Marcus-Michaelis Bousquet},
  address = {Lausanne},
  year = {1748}
}

@article{Chebyshev1854,
  author = {Chebyshev, Pafnuty},
  title = {Theorie des mecanismes connus sous le nom de parallelogrammes},
  journal = {Memoires presents a l'Academie imperiale des sciences de St.-Petersbourg},
  volume = {7},
  year = {1854},
  pages = {539--568}
}

@book{Lang1988,
  author = {Lang, Serge},
  title = {Basic Mathematics},
  publisher = {Springer},
  year = {1988}
}