Trigonometric Identities

⚡ Key Takeaways

One-line statement: Trigonometric identities are equations involving trigonometric functions that hold true for every value of the variable for which both sides are defined. Key result in words: From a single relationship — the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ — together with the addition formulas, the entire web of trigonometric identities can be derived. Where it applies: Simplifying expressions, solving trigonometric equations, evaluating integrals, modelling waves and oscillations, and rotating coordinates. Who/when: Rooted in Greek astronomy (Hipparchus, ~150 BCE) and Indian mathematics (Aryabhata, ~500 CE), formalized by Euler in the 18th century.

💡 The Core Insight

A point moving around the unit circle has coordinates $(\cos\theta, \sin\theta)$ . Because that point always lies on the circle of radius 1, its coordinates obey $x^2 + y^2 = 1$ at all times. Almost every trigonometric identity is a consequence of this geometric fact combined with how angles add when rotations are composed. Identities are the algebraic shadow of circular motion.

DefinitionTrigonometric Functions on the Unit Circle

Let $\theta$ be an angle measured counterclockwise from the positive $x$ -axis. The point where the terminal side of $\theta$ meets the unit circle has coordinates $(\cos\theta, \sin\theta)$ . The six functions are defined as:

1

\quad \sin\theta = y,\qquad \cos\theta = x,\qquad \tan\theta = \frac{\sin\theta}{\cos\theta}

Derivation hint: read the coordinates directly off the unit circle; tangent is the slope $y/x$ of the terminal ray.

2

\quad \csc\theta = \frac{1}{\sin\theta},\qquad \sec\theta = \frac{1}{\cos\theta},\qquad \cot\theta = \frac{\cos\theta}{\sin\theta}

Derivation hint: the reciprocal functions are defined as the multiplicative inverses of sine, cosine, and tangent respectively. Here $x$ and $y$ are the horizontal and vertical coordinates (dimensionless, since the radius is 1), and $\theta$ is the angle (in radians or degrees).

TheoremThe Pythagorean Identity

For every real number $\theta$ :

3

\quad \sin^2\theta + \cos^2\theta = 1

Derivation hint: apply the equation of the unit circle $x^2 + y^2 = 1$ with $x=\cos\theta$ , $y=\sin\theta$ . Dividing [3] through by $\cos^2\theta$ and by $\sin^2\theta$ yields two companions:

4

\quad 1 + \tan^2\theta = \sec^2\theta

Derivation hint: divide every term of [3] by $\cos^2\theta$ and use $\tan = \sin/\cos$ , $\sec = 1/\cos$ .

5

\quad 1 + \cot^2\theta = \csc^2\theta

Derivation hint: divide every term of [3] by $\sin^2\theta$ and use $\cot = \cos/\sin$ , $\csc = 1/\sin$ .

TheoremAngle Addition Formulas

For all angles $\alpha$ and $\beta$ :

6

\quad \sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta

Derivation hint: project a rotated unit vector onto the axes, or expand the real/imaginary parts of $e^{i(\alpha\pm\beta)} = e^{i\alpha}e^{\pm i\beta}$ .

7

\quad \cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta

Derivation hint: same as [6]; note the sign of the sine product reverses relative to the addition sign.

8

\quad \tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}

Derivation hint: divide [6] by [7] and then divide numerator and denominator by $\cos\alpha\cos\beta$ .

TheoremDouble-Angle and Half-Angle Formulas

Setting $\beta = \alpha$ in [6] and [7]:

9

\quad \sin 2\alpha = 2\sin\alpha\cos\alpha

Derivation hint: put $\beta=\alpha$ into the sine addition formula [6].

10

\quad \cos 2\alpha = \cos^2\alpha - \sin^2\alpha = 2\cos^2\alpha - 1 = 1 - 2\sin^2\alpha

Derivation hint: put $\beta=\alpha$ into [7], then substitute $\sin^2\alpha = 1-\cos^2\alpha$ (or vice versa) using [3].

11

\quad \sin^2\alpha = \frac{1-\cos 2\alpha}{2},\qquad \cos^2\alpha = \frac{1+\cos 2\alpha}{2}

Derivation hint: solve the two right-hand forms of [10] for $\sin^2\alpha$ and $\cos^2\alpha$ .

12

\quad \tan\frac{\alpha}{2} = \frac{1-\cos\alpha}{\sin\alpha} = \frac{\sin\alpha}{1+\cos\alpha}

Derivation hint: replace $\alpha$ by $\alpha/2$ in [11], take the ratio, and rationalize.

ProofDeriving the Cosine Addition Formula from Rotation

We prove $\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta$ , from which all of [6]–[8] follow. 1. Place two points on the unit circle: $P = (\cos\alpha, \sin\alpha)$ and $Q = (\cos\beta, \sin\beta)$ . 2. The squared distance between them, by the distance formula, is

13

\quad PQ^2 = (\cos\alpha - \cos\beta)^2 + (\sin\alpha - \sin\beta)^2

Derivation hint: apply $d^2 = (\Delta x)^2 + (\Delta y)^2$ to the two unit-circle points. 3. Expanding and using [3] twice (once for each angle):

14

\quad PQ^2 = 2 - 2(\cos\alpha\cos\beta + \sin\alpha\sin\beta)

Derivation hint: expand the squares, group $\cos^2+\sin^2 = 1$ for each angle, collect the cross terms. 4. Now rotate the whole picture so $Q$ sits at angle $0$ , i.e. at $(1,0)$ , and $P$ sits at angle $(\alpha - \beta)$ . The distance is unchanged by rotation, so

15

\quad PQ^2 = (\cos(\alpha-\beta) - 1)^2 + (\sin(\alpha-\beta) - 0)^2 = 2 - 2\cos(\alpha-\beta)

Derivation hint: same distance formula applied to $(\cos(\alpha-\beta),\sin(\alpha-\beta))$ and $(1,0)$ ; use [3]. 5. Equating [14] and [15] and cancelling the common $2$ :

16

\quad \cos(\alpha-\beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta\qquad \blacksquare

Derivation hint: set the two expressions for $PQ^2$ equal and divide by $-2$ . Replacing $\beta$ with $-\beta$ (and using the even/odd properties below) gives the $+$ version; the sine formulas follow from $\sin\theta = \cos(\tfrac{\pi}{2}-\theta)$ .

∎

DefinitionEven–Odd (Symmetry) Identities

Reflecting the angle through the $x$ -axis sends $\theta \to -\theta$ , flipping the sign of $y$ but not $x$ :

17

\quad \cos(-\theta) = \cos\theta,\qquad \sin(-\theta) = -\sin\theta,\qquad \tan(-\theta) = -\tan\theta

Derivation hint: the point at $-\theta$ has the same $x$ -coordinate but the negated $y$ -coordinate. Cosine is even; sine and tangent are odd.

mathematician · Leonhard Euler | 1707–1783

Euler did more than anyone to turn trigonometry from a table-bound art into algebra. His formula $e^{i\theta} = \cos\theta + i\sin\theta$ (1748) reduces every identity above to ordinary rules for exponents: the addition formulas become $e^{i\alpha}e^{i\beta} = e^{i(\alpha+\beta)}$ , and the Pythagorean identity becomes $|e^{i\theta}|^2 = 1$ . He also standardized the notation $\sin$ , $\cos$ , $\tan$ and treated the trig functions as analytic functions of a real (and complex) variable rather than as chords or half-chords of circles.

🧠 Intuition

Picture a clock hand of length 1 sweeping around. Its tip's shadow on the floor is $\cos\theta$ and its shadow on the wall is $\sin\theta$ . As it turns, those two shadows trade lengths back and forth, but the hand's length never changes — and that unchanging length, squared, is exactly $\cos^2\theta + \sin^2\theta = 1$ . The addition formulas just describe what happens to the shadows when you turn the hand by an extra angle: turning by $\alpha$ then $\beta$ is the same as turning once by $\alpha+\beta$ , so the shadow lengths must combine in a fixed bookkeeping way. Identities feel mysterious as algebra but are obvious as geometry — they are conservation laws for the spinning hand.

Figure

The unit circle: sine and cosine as coordinates of a rotating point

ExampleSimplifying an expression

Given: Simplify $\dfrac{\sin^2 x}{1 - \cos x}$ .

Steps: 1. Replace $\sin^2 x$ using the Pythagorean identity [3]: $\sin^2 x = 1 - \cos^2 x$ . 2. Factor the numerator as a difference of squares: $1 - \cos^2 x = (1-\cos x)(1+\cos x)$ . 3. The expression becomes $\dfrac{(1-\cos x)(1+\cos x)}{1-\cos x}$ . 4. Cancel the common factor $(1-\cos x)$ , valid when $\cos x \neq 1$ .

18

\quad \frac{\sin^2 x}{1-\cos x} = \boxed{\,1 + \cos x\,}\qquad (\cos x \neq 1)

Derivation hint: rewrite $\sin^2$ via [3], factor, then cancel.

ExampleComputing an exact value with the addition formula

Given: Find the exact value of $\cos 75^\circ$ .

Steps: 1. Write $75^\circ = 45^\circ + 30^\circ$ . 2. Apply the cosine addition formula [7]:

\cos 75^\circ = \cos 45^\circ\cos 30^\circ - \sin 45^\circ\sin 30^\circ

3. Substitute known values: $\cos 45^\circ = \sin 45^\circ = \tfrac{\sqrt 2}{2}$ , $\cos 30^\circ = \tfrac{\sqrt 3}{2}$ , $\sin 30^\circ = \tfrac12$ . 4. Compute:

\cos 75^\circ = \frac{\sqrt 2}{2}\cdot\frac{\sqrt 3}{2} - \frac{\sqrt 2}{2}\cdot\frac12 = \frac{\sqrt 6}{4} - \frac{\sqrt 2}{4}

19

\quad \cos 75^\circ = \boxed{\dfrac{\sqrt 6 - \sqrt 2}{4} \approx 0.2588}

Derivation hint: split 75° into known special angles and apply [7].

⚠️ Common mistake

The function notation is not distributive. In general

20

\quad \sin(\alpha + \beta) \neq \sin\alpha + \sin\beta

Derivation hint: test $\alpha=\beta=90^\circ$ : left side $=\sin 180^\circ = 0$ , right side $= 1+1 = 2$ . Also be careful: $\sin^2\theta$ means $(\sin\theta)^2$ , while $\sin^{-1}\theta$ means the inverse (arcsine), not $1/\sin\theta = \csc\theta$ . And every identity carries hidden domain restrictions — cancellations like the one in [18] are only valid where the cancelled factor is nonzero.

Counterexample

Students often "prove" an identity by manipulating both sides until they reach $0=0$ , then declare victory. But this can validate a false statement if a step is reversible only conditionally. For instance, squaring is not reversible: from the false equation $\sin x = -\sin x$ one can square to get the true $\sin^2 x = \sin^2 x$ . The squaring step introduced solutions that were not in the original. Rule: to prove an identity, transform one side into the other using only equivalences; never assume the thing you are trying to prove.

connection

Trigonometric identities sit at a crossroads of several fields. They are the algebraic backbone of Solving Trigonometric Equations, where rewriting an expression into a single function often unlocks the solution. They power Trigonometric Integration, where formulas like [11] convert un-integrable powers into linear combinations of cosines. They are equivalent statements about Complex Numbers via Euler's formula, turning trigonometry into the arithmetic of points on the unit circle in the complex plane. They generalize the Pythagorean Theorem (set $\theta$ to recover $a^2+b^2=c^2$ in a right triangle), and they underlie Fourier Series, where every periodic signal is built from sines and cosines that must be combined using these very identities.

🔬 Where this appears

⏳ History

- ~150 BCE — Hipparchus of Nicaea compiles the first known table of chords, the direct ancestor of the sine function, for astronomical prediction.

- ~150 CE — Ptolemy in the Almagest states "Ptolemy's theorem," which is geometrically equivalent to the sine and cosine addition formulas.

- ~500 CE — Aryabhata (India) tabulates the jya (half-chord), introducing what would become the sine; the word "sine" descends from a mistranslation of this term.

- ~10th century — Abu al-Wafa al-Buzjani (Persia) derives the double-angle and addition formulas in essentially modern form and introduces the tangent.

- 1464 — Regiomontanus writes De triangulis omnimodis, the first European treatise treating trigonometry as an independent subject.

- 1614–1620 — John Napier & Henry Briggs publish logarithms, making the product-to-sum identities a practical computing shortcut for centuries.

- 1748 — Leonhard Euler publishes Introductio in analysin infinitorum, giving

e^{i\theta}=\cos\theta+i\sin\theta

and modern functional notation, unifying all identities.

- 1822 — Joseph Fourier shows arbitrary periodic functions decompose into sines and cosines, vastly extending the relevance of the identities.

open-problem

Identities are clean over the real numbers, but extending them raises genuine subtleties. When trig functions are continued to complex arguments, $\sin z$ and $\cos z$ become unbounded (e.g. $\cos(iy) = \cosh y \to \infty$ ), yet the addition formulas [6]–[7] still hold formally — a reminder that "identity" depends on domain. The inverse functions are not single-valued: $\arcsin$ requires a branch choice, and "solving" $\sin x = c$ yields infinitely many solutions, so an "identity" in inverse functions like $\arcsin(\sin x) = x$ holds only on a restricted interval. At the research frontier, questions about which sums of trig values are rational or algebraic (Niven's theorem and its generalizations) remain delicate, and the transcendence of values like $\cos(1)$ connects to deep open problems in number theory.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also