Trigonometric identities: sum, difference, and double-angle

Intuition [Beginner]

The sum and difference formulas tell you the sine and cosine of $A+B$ in terms of sines and cosines of $A$ and $B$. They are:

$$\cos (A+B)=\cos A\cos B-\sin A\sin B$$ $$\sin (A+B)=\sin A\cos B+\cos A\sin B$$

Setting $A=B$ gives the double-angle formulas:

$$\cos (2A)={\cos }^{2}A-{\sin }^{2}A$$ $$\sin (2A)=2\sin A\cos A$$

These formulas convert an angle you have not measured into combinations of angles you know. If you know $\sin 30^{\circ }=0.5$ and $\cos 45^{\circ }=\sqrt{2}/2$, you can compute $\sin 75^{\circ }$ without a calculator.

The key insight is that rotating by $A+B$ is the same as rotating by $A$ and then by $B$. The formulas come from multiplying two rotation matrices.

Visual [Beginner]

A unit circle with two angles $A$ and $B$ shown. The point at angle $A+B$ has coordinates $(\cos (A+B),\sin (A+B))$. Arrows show the composition of two rotations.

The sum formula: rotating by two angles in sequence is the same as rotating by their sum.

Worked example [Beginner]

Compute $\sin 75^{\circ }$ using the sum formula.

Write $75^{\circ }=45^{\circ }+30^{\circ }$. Then:

$$\sin 75^{\circ }=\sin 45^{\circ }\cos 30^{\circ }+\cos 45^{\circ }\sin 30^{\circ }$$ $$=\frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\cdot \frac{1}{2}$$ $$=\frac{\sqrt{6}+\sqrt{2}}{4}\approx 0.9659$$

This is exact and avoids any rounding from a calculator.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Theorem (Sum and difference formulas). For all angles $A$ and $B$:

$$\sin (A\pm B)=\sin A\cos B\pm \cos A\sin B$$ $$\cos (A\pm B)=\cos A\cos B\mp \sin A\sin B$$

Corollary (Double-angle formulas). Setting $B=A$:

$$\sin (2A)=2\sin A\cos A$$ $$\cos (2A)={\cos }^{2}A-{\sin }^{2}A=2{\cos }^{2}A-1=1-2{\sin }^{2}A$$

Corollary (Half-angle formulas). Solving the double-angle cosine for $\cos (A/2)$:

$${\cos }^2(A/2)=\frac{1+\cos A}{2},{\sin }^2(A/2)=\frac{1-\cos A}{2}$$

Key theorem with proof [Intermediate+]

Theorem (Sum formula via rotation matrices). $\cos (A+B)=\cos A\cos B-\sin A\sin B$ and $\sin (A+B)=\sin A\cos B+\cos A\sin B$.

Proof. A rotation by angle $\theta$ in the plane is represented by the matrix:

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

Rotating by $A+B$ is the same as rotating by $A$ and then by $B$:

$$R(A+B)=R(B)\cdot R(A).$$

Computing the matrix product:

$$R(B)\cdot R(A)=\left(\begin{array}{cc}\cos B& -\sin B\\ \sin B& \cos B\end{array}\right)\left(\begin{array}{cc}\cos A& -\sin A\\ \sin A& \cos A\end{array}\right)$$

$$=\left(\begin{array}{cc}\cos B\cos A-\sin B\sin A& -\cos B\sin A-\sin B\cos A\\ \sin B\cos A+\cos B\sin A& -\sin B\sin A+\cos B\cos A\end{array}\right).$$

Comparing entries with $R(A+B)=\left(\begin{array}{cc}\cos (A+B)& -\sin (A+B)\\ \sin (A+B)& \cos (A+B)\end{array}\right)$ gives the formulas. $\square$

Bridge. The proof connects trigonometry to linear algebra through rotation matrices; the foundational reason the formulas work is that the composition of two rotations is a rotation by the sum of the angles. This pattern appears again in the theory of complex numbers where $e^{i(A+B)}=e^{iA}{e}^{iB}$ encodes the same formulas via Euler's formula. The bridge is that the sum formulas are the group-law expression for the circle group $SO(2)$.

Exercises [Intermediate+]

Advanced results [Master]

Product-to-sum identities. Every product of two trigonometric functions can be written as a sum:

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

These are the key tool in Fourier analysis: multiplying two Fourier modes produces two new frequencies (the sum and difference frequencies), which is the algebraic content of convolution.

The Weierstrass substitution. The substitution $t=\tan (\theta /2)$ converts any rational function $R(\sin \theta ,\cos \theta )$ into a rational function of $t$, which can then be integrated by partial fractions. This universal method for integrating $R(\sin x,\cos x)$ was central to 19th-century calculus.

Chebyshev polynomials. The identity $\cos (n\theta )=T_n(\cos \theta )$ defines the Chebyshev polynomial $T_n$. The double-angle formula gives $T_2(x)=2x^2-1$, and the sum formula yields the recurrence $T_{n+1}(x)=2x{T}_n(x)-T_{n-1}(x)$.

Synthesis. The sum and difference formulas are the algebraic expression of the circle group structure; the central insight is that $e^{i\theta }$ converts addition of angles into multiplication of complex numbers, making trigonometric identities into polynomial identities. This pattern appears again in the Chebyshev recurrence and in the Fourier convolution theorem. The bridge is that the sum formulas generalise the addition law of angles to a multiplication law for complex exponentials — this builds toward the representation theory of $U(1)$ 07.01.09 and the harmonic analysis of periodic functions.

Full proof set [Master]

Proposition (Euler's formula implies the sum formulas). From $e^{i\theta }=\cos \theta +i\sin \theta$ and the law of exponents $e^{i(A+B)}=e^{iA}{e}^{iB}$, the sum formulas for sine and cosine follow immediately.

Proof. Expand:

$$e^{i(A+B)}=\cos (A+B)+i\sin (A+B)$$ $$e^{iA}{e}^{iB}=(\cos A+i\sin A)(\cos B+i\sin B)=(\cos A\cos B-\sin A\sin B)+i(\sin A\cos B+\cos A\sin B).$$

Equating real and imaginary parts gives the sum formulas. $\square$

Connections [Master]

Unit circle trigonometry 00.07.01 provides the geometric setting where angles correspond to points on the circle; the sum formulas describe how composition of rotations acts on these points.

The law of sines and cosines 00.08.02 applies the sum formulas to derive the triangle laws, which are themselves special cases of the rotation-group structure.

Euler's formula $e^{i\theta }=\cos \theta +i\sin \theta$ encodes the sum formulas as the multiplicative property of complex exponentials, connecting trigonometry to complex analysis 06.01.01.

Bibliography [Master]

@book{stewart-precalculus,
  author = {Stewart, James and Redlin, Lothar and Watson, Saleem},
  title = {Precalculus: Mathematics for Calculus},
  edition = {8},
  publisher = {Cengage},
  year = {2019}
}

@book{apostol-calculus-v1,
  author = {Apostol, Tom M.},
  title = {Calculus},
  volume = {1},
  edition = {2},
  publisher = {Wiley},
  year = {1967}
}

@book{euler-introductio,
  author = {Euler, Leonhard},
  title = {Introductio in analysin infinitorum},
  year = {1748},
  note = {Vol. 1 Ch. 8--9; translated by Blanton 1988}
}

@incollection{ptolemy-almagest,
  author = {Ptolemy, Claudius},
  title = {Almagest},
  year = {c. 150 AD},
  note = {Book I Ch. 10--11; chord addition formula}
}

Historical & philosophical context [Master]

The sum and difference formulas have ancient roots. Ptolemy's chord addition formula in the Almagest ^{[Ptolemy ~150 AD]} is equivalent to $\sin (A+B)$ expressed in terms of chords. The modern algebraic form emerged with Euler's systematic treatment in the Introductio (1748) ^{[Euler 1748]}, where the connection to complex exponentials was made explicit.

The double-angle formulas are essential in calculus: they appear in the differentiation of trigonometric functions (via the limit definition), in integration (e.g., $\int {\sin }^{2}xdx=\int \frac{1-\cos 2x}{2}dx$), and in Fourier analysis where they express energy conservation. The product-to-sum identities are the algebraic content of the Fourier convolution theorem, making these formulas foundational for signal processing.