Trigonometric Identities: Sum, Double-Angle Formulas and Proofs

What makes an equation a trigonometric identity?

A trigonometric identity is an equation that is true for every angle at which both of its sides are defined; an ordinary equation is true only for some angles. \sin x=\frac12 is an equation: it is true at 30^\circ and 150^\circ and false at 45^\circ. \sin^2x+\cos^2x=1 is an identity: it is true at 45^\circ, at 1000^\circ and at every other angle. Solving an equation means finding the angles that make it true; proving an identity means showing that all of them do.

Two details are worth knowing from the start.

• The sides need not exist everywhere. \tan x=(\sin x)/(\cos x) is an identity although neither side exists at 90^\circ, because the claim is only about the angles where both do. Cancelling can change where an expression is defined: (\sin x\cos x)/(\cos x) is not defined at 90^\circ but \sin x is, so the two are equal wherever both are defined, which is as much as an identity ever asks.
• A false formula needs only one counterexample. \sin(x+y)=\sin x+\sin y looks reasonable, but at x=y=30^\circ the left side is \sin60^\circ≈0.866 and the right side is 0.5+0.5=1. A true identity, on the other hand, can never be shown by trying angles, since there are infinitely many; it has to be proved.

This article lists the identities worth knowing, shows where they come from (everything follows from the unit circle and one rotation), and then shows how to prove one and how to test a claim before spending time on it. The basics of the sine, cosine and tangent are in the article on basic trigonometry; here are the proofs and the longer list.

Where do the fundamental trigonometric identities come from?

The fundamental identities are the reciprocal, quotient, Pythagorean, cofunction, even-odd and shift identities, and every one of them can be read straight off the unit circle. On the unit circle the point at the angle x is (\cos x,\sin x), which is where the sine and cosine come from for every angle (see the unit circle).

Reciprocal and quotient. These are the definitions of \tan, \cot, \sec and \csc in terms of \sin and \cos:

Pythagorean. The point is on a circle of radius 1, so its coordinates satisfy x^2+y^2=1, which is \cos^2x+\sin^2x=1. Divide it by \cos^2x and you get 1+\tan^2x=\sec^2x; divide it by \sin^2x and you get 1+\cot^2x=\csc^2x. It is the Pythagorean theorem in disguise.

Cofunction. The two acute angles of a right triangle add up to 90^\circ, and the side that is opposite one is adjacent to the other, so the sine of an angle is the cosine of its complement.

Even and odd. Turning by -x mirrors the point in the x-axis: the x-coordinate stays and the y-coordinate changes sign.

Shifts by 90^\circ and 180^\circ. A half turn sends (x,y) to (-x,-y), and a quarter turn sends it to (-y,x).

† An identity that holds wherever both sides are defined: the two sides are not defined at the same angles.

Every formula in the tables on this page was tested by a program at hundreds of angles, against an independent calculation, and the ones with a dagger are exactly the ones where one side is defined and the other is not at some angle.

How do the sum and difference formulas work?

The sum formulas come from turning a point through an angle: the point at x, turned through y, lands at the angle x+y, and its new coordinates are made of the four products \cos x\cos y, \sin x\sin y, \sin x\cos y and \cos x\sin y.

Turn the whole plane about the origin through the angle y. The point (1,0) goes to (\cos y,\sin y) and the point (0,1) goes to (-\sin y,\cos y). A rotation keeps sums and multiples, and the point at the angle x is \cos x·(1,0)+\sin x·(0,1), so it goes to

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

=\left(\cos x\cos y-\sin x\sin y, \sin x\cos y+\cos x\sin y\right).

But turning the point at x through y puts it at the angle x+y, whose coordinates are (\cos(x+y),\sin(x+y)). Comparing the two coordinates gives both formulas at once.

† The tangent formulas hold wherever both sides are defined.

The difference formulas are the sum formulas with y replaced by -y, using \cos(-y)=\cos y and \sin(-y)=-\sin y, which flips one sign in each: look at the minus in \cos(x+y)=\cos x\cos y-\sin x\sin y and the plus in \cos(x-y)=\cos x\cos y+\sin x\sin y. The tangent formulas are the quotient of the two: divide the numerator and the denominator of (\sin(x+y))/(\cos(x+y)) by \cos x\cos y.

Move the angles below and watch the four products make up the new point.

How do you get the double-angle, half-angle and power-reducing formulas?

Put y=x in the sum formulas and you get the double-angle formulas; solve those for the squares and you get the half-angle and power-reducing formulas. Nothing new is needed: it is the same rotation.

Double angle. \sin2x=\sin(x+x)=2\sin x\cos x and \cos2x=\cos^2x-\sin^2x. Because \sin^2x+\cos^2x=1, the cosine has three forms: replace \sin^2x by 1-\cos^2x and you get 2\cos^2x-1; replace \cos^2x by 1-\sin^2x and you get 1-2\sin^2x. Which one to use depends on what you want to be left with.

Triple angle. \sin3x=\sin(2x+x) and \cos3x=\cos(2x+x): use the sum formula, then the double-angle ones, then \sin^2x+\cos^2x=1.

Half angle. Take \cos x=1-2\sin^2x/2 and \cos x=2\cos^2x/2-1 (the double-angle formulas for the angle x/2) and solve for the squares:

Taking the square root gives \sinx/2=±√((1-\cos x)/(2)) and \cosx/2=±√((1+\cos x)/(2)), and the sign is the sign of the sine or cosine in the quadrant where x/2 lies, not a free choice. Power reducing. The same two equations with x in place of x/2 turn a square into a first power of a double angle, which is what an integral or a signal analysis wants:

Exact values. With these formulas the angles of 30^\circ, 45^\circ and 60^\circ lead to many more. \sin15^\circ=\sin(45^\circ-30^\circ), which is (√2)/(2)·(√3)/(2)-(√2)/(2)·\frac12, that is (√6-√2)/(4). \sin22.5^\circ is half of 45^\circ: \sin^2 22.5^\circ=(1-\cos45^\circ)/(2)=(2-√2)/(4), so \sin22.5^\circ=(√(2-√2))/(2). Even the pentagon's angles (18^\circ, 36^\circ, 54^\circ, 72^\circ) are exact: \cos36^\circ=(1+√5)/(4), half the golden ratio. Pick any of them below to see the derivation.

What are the product-to-sum formulas, and what is R sin(x + φ)?

Adding and subtracting the sum and difference formulas turns a product of sines and cosines into a sum, and the reverse turns a sum into a product. Add \sin(x+y)=\sin x\cos y+\cos x\sin y and \sin(x-y)=\sin x\cos y-\cos x\sin y: the second terms cancel and \sin(x+y)+\sin(x-y)=2\sin x\cos y. The other three come the same way.

Sum to product. Write the angles as x=(u+v)/(2)+(u-v)/(2) and y=(u+v)/(2)-(u-v)/(2) in the product formulas, and a sum of two sines or cosines becomes a product:

One wave from two. The sum of a sine and a cosine of the same angle is again a single wave. Look for numbers R and φ with R\cosφ=a and R\sinφ=b: then

a\sin x+b\cos x=R\sin x\cosφ+R\cos x\sinφ=R\sin(x+φ),

by the sum formula. Squaring and adding the two conditions gives R^2=a^2+b^2, so R=√(a^2+b^2), and φ=\operatornameatan2(b,a). For example 3\sin x+4\cos x=5\sin(x+53.13^\circ), because R=√(9+16)=5.

This does three jobs. The largest and smallest values of a\sin x+b\cos x are ± R (here ±5). An equation such as 3\sin x+4\cos x=2.5 becomes the basic equation \sin(x+53.13^\circ)=0.5, whose solutions are x+53.13^\circ=30^\circ or 150^\circ (plus whole turns), so x≈336.87^\circ or 96.87^\circ between 0^\circ and 360^\circ. Two waves of the same frequency add up to another wave of that frequency, which is how alternating currents and sound waves combine. Try other values of a and b below.

How do you prove a trigonometric identity?

To prove an identity, start from one side and change it, one valid step at a time, until it becomes the other side. Every line must equal the line before it, and the last line must be the other side of the claim.

1. Pick a side. Usually the more complicated one, because simplifying is easier than growing.
2. Do not move terms across the equals sign and do not multiply or add to both sides. That assumes what you want to prove. (You may work on both sides separately until they meet in the middle.)
3. Turn everything into sine and cosine when you do not see a path: \tan, \cot, \sec, \csc all have a definition in \sin and \cos.
4. Use the algebra you know: common denominators, expanding, factoring, the difference of squares a^2-b^2=(a-b)(a+b).
5. Use \sin^2x+\cos^2x=1 in whichever direction helps: 1→\sin^2x+\cos^2x, 1-\cos^2x→\sin^2x.
6. Reach for the sum, double-angle and half-angle formulas when an angle such as 2x, x+y or x/2 appears.
7. Say where it holds: wherever both sides are defined.

Example 1. Prove \tan x+\cot x=\sec x\csc x. Start from the left side. \tan x+\cot x=(\sin x)/(\cos x)+(\cos x)/(\sin x) (quotient identities) =(\sin^2x+\cos^2x)/(\sin x\cos x) (common denominator) =(1)/(\sin x\cos x) (Pythagorean) =\sec x\csc x (reciprocals). Done.

Example 2. Prove (1-\cos x)(1+\cos x)=\sin^2x. Start from the left side. It equals 1-\cos^2x (difference of squares), which is \sin^2x (Pythagorean). Done.

The nine proofs below go from these first ones to the triple angle and the half-angle tangent. Each line carries its reason, and a program has checked that each line equals the one before it.

How can you test an identity before proving it, and what can a test not tell you?

Test a claimed identity by comparing both sides at many angles: a single difference refutes it, but agreement, however long, only makes it likely. This saves a great deal of time, because trying to prove a false statement is a trap. Evaluate both sides at a few angles that are not special (25^\circ, 40^\circ, 100^\circ) and not too symmetric. If the sides differ anywhere, you have a counterexample and are done.

What a test cannot do is prove. \sin2x=2\sin x is true at x=0^\circ and x=180^\circ, and a test at those two angles alone would pass it; it fails at 30^\circ, where \sin60^\circ≈0.866 and 2\sin30^\circ=1. Even a test at a thousand angles is only evidence. An identity is proved with algebra or geometry, as in the previous section.

A test also has to deal with two practical matters:

• Undefined points. If one side is undefined where the other is not (cancelling \cos x removes 90^\circ from the domain), the sides are equal where both are defined, and the tester below reports that case separately instead of calling it a difference.
• Rounding. A computer does not get \sin^2x+\cos^2x=1 exactly: the difference is about 10^-16, so compare with a small tolerance, never with equality. Near 90^\circ a tangent is huge, and a difference of two huge numbers loses its digits, so a sensible tester skips points where the numbers are too large.

Type your own claim below, in degrees, with x (and y for two angles). It compares the sides at every whole degree from 0^\circ to 359^\circ and at hundreds of other angles.

Is there one rule behind all of these formulas?

Yes: Euler's formula e^iθ=\cosθ+i\sinθ contains the sum formulas, and with them everything else in this article. The exponential has the rule e^ix· e^iy=e^i(x+y). Write each factor with Euler's formula and multiply out, remembering i^2=-1:

(\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).

The left side is e^i(x+y)=\cos(x+y)+i\sin(x+y). Two complex numbers are equal only if their real parts and their imaginary parts are equal, so the real parts give the cosine sum formula and the imaginary parts give the sine sum formula. It is the rotation of the previous section, since multiplying by e^iy is turning through y.

Raising to a power gives de Moivre's formula, (\cos x+i\sin x)^n=\cos nx+i\sin nx. For n=3, expanding the left side gives \cos3x=\cos^3x-3\cos x\sin^2x and \sin3x=3\cos^2x\sin x-\sin^3x, which are the triple-angle formulas written with both functions (use \sin^2x+\cos^2x=1 to get the forms above). Exactly one rule about exponents sits behind the whole list.

How do you check an identity in a program?

A program checks an identity by sampling: evaluate both sides at many angles and report the largest difference, which should be around 10^-16 for a true identity and large for a false one. Three details matter. Convert degrees to radians before calling sin and cos. Compare with a tolerance such as 10^-12 rather than ==. And skip the angles where a side is undefined (tan at 90^\circ or a division by zero) rather than calling them differences.

The programs below measure the largest gap for four claims: \sin^2x+\cos^2x=1, \sin2x=2\sin x\cos x, the false \sin2x=2\sin x, and 3\sin x+4\cos x=5\sin(x+53.13^\circ). They print whether the gap is below the tolerance (true for the identities), and for the false one the size of the gap and where it is largest (2.5981 at 120^\circ). For symbolic proofs of the same statements, use a computer algebra system such as SymPy: it can simplify one side into the other, which a numeric sample never can.

import math


def max_gap(f, g, n=3600):
    """The largest |f(x) - g(x)| over n + 1 angles from 0 to 360 degrees, and where it first happens."""
    worst, at = 0.0, 0.0
    for k in range(n + 1):
        x = 360 * k / n
        try:
            gap = abs(f(x) - g(x))
        except ZeroDivisionError:
            continue                    # undefined here: skip the point, do not call it a difference
        if gap > worst:
            worst, at = gap, x
    return worst, at


r = math.radians
sin = lambda x: math.sin(r(x))
cos = lambda x: math.cos(r(x))

print(max_gap(lambda x: sin(x) ** 2 + cos(x) ** 2, lambda x: 1)[0] < 1e-12)                          # True
print(max_gap(lambda x: sin(2 * x), lambda x: 2 * sin(x) * cos(x))[0] < 1e-12)                       # True
print(max_gap(lambda x: sin(x + 25), lambda x: sin(x) * cos(25) + cos(x) * sin(25))[0] < 1e-12)      # True
gap, at = max_gap(lambda x: sin(2 * x), lambda x: 2 * sin(x))
print(round(gap, 4), at)                                                                              # 2.5981 120.0: not an identity
print(round(max_gap(lambda x: sin(x + 25), lambda x: sin(x) + sin(25))[0], 4))                       # 0.8555: not an identity either
R = math.hypot(3, 4)
phi = math.degrees(math.atan2(4, 3))
print(round(R, 4), round(phi, 4))                                                                    # 5.0 53.1301
print(max_gap(lambda x: 3 * sin(x) + 4 * cos(x), lambda x: R * sin(x + phi))[0] < 1e-12)            # True

const r = (d) => (d * Math.PI) / 180
const sin = (x) => Math.sin(r(x))
const cos = (x) => Math.cos(r(x))

function maxGap(f, g, n = 3600) {
  // the largest |f(x) - g(x)| over n + 1 angles from 0 to 360 degrees, and where it first happens
  let worst = 0, at = 0
  for (let k = 0; k <= n; k++) {
    const x = (360 * k) / n
    const gap = Math.abs(f(x) - g(x))
    if (Number.isNaN(gap)) continue            // undefined here: skip the point, do not call it a difference
    if (gap > worst) { worst = gap; at = x }
  }
  return [worst, at]
}

maxGap((x) => sin(x) ** 2 + cos(x) ** 2, () => 1)[0] < 1e-12                                  // true
maxGap((x) => sin(2 * x), (x) => 2 * sin(x) * cos(x))[0] < 1e-12                               // true
maxGap((x) => sin(x + 25), (x) => sin(x) * cos(25) + cos(x) * sin(25))[0] < 1e-12              // true
const [gap, at] = maxGap((x) => sin(2 * x), (x) => 2 * sin(x))
;[+gap.toFixed(4), at]                                                                          // [2.5981, 120]: not an identity
+maxGap((x) => sin(x + 25), (x) => sin(x) + sin(25))[0].toFixed(4)                              // 0.8555: not an identity either
const R = Math.hypot(3, 4), phi = (Math.atan2(4, 3) * 180) / Math.PI
;[+R.toFixed(4), +phi.toFixed(4)]                                                               // [5, 53.1301]
maxGap((x) => 3 * sin(x) + 4 * cos(x), (x) => R * sin(x + phi))[0] < 1e-12                      // true

#include <iostream>
#include <cstdio>
#include <cmath>
using namespace std;

const double PI = 3.14159265358979323846;
double worstAt;                                  // where the largest gap is

double rad(double d) { return d * PI / 180.0; }

// the two sides of claim number "which", at x degrees
double lhs(int which, double x) {
  double a = rad(x);
  if (which == 1) return sin(a) * sin(a) + cos(a) * cos(a);
  if (which == 2) return sin(2 * a);
  if (which == 3) return sin(2 * a);
  return 3 * sin(a) + 4 * cos(a);
}

double rhs(int which, double x) {
  double a = rad(x);
  if (which == 1) return 1;
  if (which == 2) return 2 * sin(a) * cos(a);
  if (which == 3) return 2 * sin(a);
  return 5 * sin(a + rad(53.13010235415598));
}

double maxGap(int which) {
  double worst = 0;
  worstAt = 0;
  for (int k = 0; k <= 3600; k++) {
    double x = 360.0 * k / 3600.0;
    double gap = fabs(lhs(which, x) - rhs(which, x));
    if (gap > worst) { worst = gap; worstAt = x; }
  }
  return worst;
}

int main() {
  printf("%d\n", maxGap(1) < 1e-12 ? 1 : 0);
  printf("%d\n", maxGap(2) < 1e-12 ? 1 : 0);
  printf("%.4f %.1f\n", maxGap(3), worstAt);
  printf("%d\n", maxGap(4) < 1e-9 ? 1 : 0);
  return 0;
}

/* output:
1
1
2.5981 120.0
1
*/

Where do the trigonometric identities come from?

The sum formulas began as a theorem about chords in a circle, became tools for multiplying large numbers before logarithms, and were folded into one rule about exponentials in the eighteenth century.

• About 150 AD. Ptolemy's Almagest proves a theorem about the sides and diagonals of a quadrilateral inscribed in a circle. It gives the chord of a sum or a difference of two arcs, which is the sum formula for the sine in the language of chords, and Ptolemy used it to build his table.
• About 500 AD. Indian astronomers state the rules for the sine and cosine of sums, and for the half angle, as part of their sine tables.
• 16th century. Prosthaphaeresis: before logarithms, astronomers (Tycho Brahe's circle among them) turned a multiplication into an addition with \cos x\cos y=(\cos(x-y)+\cos(x+y))/(2), looking the three cosines up in a table. Napier's logarithms (1614) replaced the method.
• 1722. de Moivre's formula, the form (\cos x+i\sin x)^n=\cos nx+i\sin nx, ties the multiple-angle formulas to powers.
• 1748. Euler's Introductio makes the modern notation standard and gives e^ix=\cos x+i\sin x, from which the sum formulas follow in two lines.

What are the common mistakes with trigonometric identities?

The most common mistakes with trigonometric identities are the nine below, each with the correct statement.

Practice: test your understanding

These questions mix everything above. A wrong answer costs nothing here: read the hint and try again.

Summary: trigonometric identities at a glance

• Identity: true wherever both sides are defined; a false formula needs one counterexample, a true one needs a proof.
• Fundamental: \sin^2x+\cos^2x=1, 1+\tan^2x=\sec^2x, \tan x=(\sin x)/(\cos x), \sin(90^\circ-x)=\cos x, \sin(-x)=-\sin x, \cos(-x)=\cos x.
• Sum and difference: \sin(x± y)=\sin x\cos y±\cos x\sin y, \cos(x± y)=\cos x\cos y\mp\sin x\sin y, from turning a point through y.
• Double and half: \sin2x=2\sin x\cos x; \cos2x=\cos^2x-\sin^2x, also 2\cos^2x-1 and 1-2\sin^2x; \sin^2x/2=(1-\cos x)/(2), \cos^2x/2=(1+\cos x)/(2).
• Product and sum: \sin x\cos y=(\sin(x+y)+\sin(x-y))/(2) and the other three; sums become products with the half sum and half difference.
• One wave: a\sin x+b\cos x=R\sin(x+φ), R=√(a^2+b^2), φ=\operatornameatan2(b,a).
• Proving: one side only, valid steps, sine and cosine when stuck, Pythagorean in either direction, state the domain.
• Testing: compare at many angles, tolerance not equality, skip undefined points; a test refutes but never proves.
• Behind it all: e^ix· e^iy=e^i(x+y).

Glossary: key terms

Trigonometric identity

An equation between trigonometric expressions that is true for every angle at which both sides are defined.

Counterexample

One angle at which the two sides of a claimed identity differ, which proves the claim false.

Pythagorean identity

The identity sin squared plus cos squared equals 1, which says that the point at an angle on the unit circle is at distance 1 from the center.

Sum formula

A formula that gives the sine, cosine or tangent of x plus y from the functions of x and of y, such as sin(x + y) = sin x cos y + cos x sin y.

Double-angle formula

A formula for the sine, cosine or tangent of 2x in terms of the functions of x, found by putting y equal to x in the sum formula.

Half-angle formula

A formula for the sine, cosine or tangent of x over 2, with a square root whose sign is that of the function in the quadrant of x over 2.

Product-to-sum formula

A formula that writes a product of two sines or cosines as half a sum, such as sin x cos y = (sin(x + y) + sin(x - y)) / 2.

Single-wave form

The rewriting of a sin x + b cos x as R sin(x + phi) with R equal to the square root of a squared plus b squared.

Domain of an identity

The angles at which both sides are defined; an identity says nothing about angles where a side is undefined.

Prosthaphaeresis

A sixteenth-century method that used the product-to-sum formulas to replace a multiplication by additions and table look-ups.

Frequently asked questions

References

1. Milton Abramowitz, Irene A. Stegun (1964). Handbook of Mathematical Functions (section 4.3, circular functions). National Bureau of Standards.
2. Eli Maor (1998). Trigonometric Delights. Princeton University Press.
3. I. M. Gelfand, Mark Saul (2001). Trigonometry (2nd ed.). Birkhäuser.
4. James Stewart, Lothar Redlin, Saleem Watson (2016). Precalculus: Mathematics for Calculus (chapters on analytic trigonometry). Cengage Learning.
5. G. J. Toomer (1984). Ptolemy's Almagest (translated and annotated). Springer.
6. Carl B. Boyer, Uta C. Merzbach (2011). A History of Mathematics (3rd ed.). Wiley.
7. Leonhard Euler (1748). Introductio in analysin infinitorum. Marcum-Michaelem Bousquet.

Related articles

• Basic Trigonometry: Sine, Cosine and Tangent: Basic trigonometry explained: sine, cosine and tangent of a right triangle, exact values at 30°, 45° and 60°, the unit circle, identities, equations and graphs.
• Angles: Measuring, Types, Pairs and Parallel Lines: Angles explained: degrees, radians and turns, types and pairs of angles, parallel lines and transversals, polygon angles, clock angles and bearings.
• Triangles: Types, Angles, Area, Centers and Congruence: Triangles explained: types by sides and angles, the 180° angle sum, the triangle inequality, area and Heron's formula, centers, congruence and the law of sines.
• Derivatives of Trigonometric Functions: Derivatives of trigonometric functions with proofs: sin x and cos x, then tan, cot, sec and csc, why radians, simple harmonic motion, higher derivatives, and code.