Fundamental trigonometric identities

Summary

This lesson opens analytic trigonometry, which studies the algebraic relations among the trigonometric functions. It defines precisely what an identity is, an equality that holds on the whole common domain of its two sides, and gathers the fundamental identities: the reciprocal, quotient, Pythagorean, parity and cofunction identities, which can also be read as lengths of line segments in the unit circle. It teaches the method for proving an identity, transforming one side into the other or both into the same expression, and shows why it is not valid to start from the identity and square it; a single exception, a counterexample, suffices to refute a false identity. It expresses each function in terms of another, with the sign fixed by the quadrant, simplifies expressions while keeping track of the domain, converts algebraic roots into trigonometric functions by means of substitutions and, by eliminating the parameter, recognizes in the point $\left( a \cos t, b \sin t \right)$ an ellipse and in the point $\left( a \sec t, b \tan t \right)$ a hyperbola.

Learning objectives

Define a trigonometric identity precisely, determine its common domain and distinguish it from an equation, and gather and prove the fundamental identities, also in their geometric reading as line segments.
Prove identities by transforming one side into the other or both into a common expression, recognize invalid reasoning and refute a false identity with a counterexample.
Express each trigonometric function in terms of another, with the sign that corresponds to the quadrant, and simplify expressions while preserving their domain.
Use trigonometric substitutions to transform expressions with roots, and eliminate the parameter in the trigonometric parametrizations of the ellipse and of the hyperbola.

What an identity is

Some equalities between trigonometric functions hold for every real number, such as $\sin^2 \theta + \cos^2 \theta = 1$ , proved in the lesson on trigonometric ratios and the unit circle; others, such as $\sin \theta = -\frac{1}{2}$ , hold only for some values, which the lesson on the inverse trigonometric functions taught us to find. The difference is one of nature: the first asserts something about all numbers, and the second poses a question. Now, the phrase “for every real number” calls for a refinement, because many trigonometric expressions are not defined at all numbers.

A trigonometric identity is an equality between two expressions built from trigonometric functions that holds for all values of the variable at which both sides are defined. The set of those values is the common domain of the identity. An equality that holds only for some values of the common domain is an equation, and those values are its solutions.

The domain is not a detail. The tangent is defined where the cosine does not vanish, and the cotangent where the sine does not vanish; therefore the product of the two is defined exactly when $x$ is not an integer multiple of $\frac{\pi}{2}$ , and there it equals $1$ :

\tan x \cot x = 1, \qquad x \neq \frac{k\pi}{2}

At the multiples of $\frac{\pi}{2}$ the identity asserts nothing, because its left-hand side denotes no number. Likewise, $\frac{\sin x}{\tan x} = \cos x$ is an identity on its common domain, which excludes the multiples of $\frac{\pi}{2}$ ; however, the functions $f(x) = \frac{\sin x}{\tan x}$ and $g(x) = \cos x$ are not equal, because by the Course I lesson on function operations two equal functions have the same domain, and the second is defined, for example, at $x = 0$ . An identity, therefore, compares two expressions only where both make sense.

The fundamental identities

The following identities are obtained directly from the definitions and from the theorems of the preceding unit; all the others are deduced from them.

TheoremFundamental identities

On the common domain of each equality, the following reciprocal, quotient and Pythagorean identities hold. Moreover, the sine, the tangent, the cosecant and the cotangent are odd functions, and the cosine and the secant are even functions; and each function of $\frac{\pi}{2} - \theta$ is its cofunction of $\theta$ , where the sine and the cosine, the tangent and the cotangent, the secant and the cosecant are cofunctions of one another.

\sin \theta \csc \theta = 1, \qquad \cos \theta \sec \theta = 1, \qquad \tan \theta \cot \theta = 1

\tan \theta = \frac{\sin \theta}{\cos \theta}, \qquad \cot \theta = \frac{\cos \theta}{\sin \theta}

\sin^2 \theta + \cos^2 \theta = 1, \qquad 1 + \tan^2 \theta = \sec^2 \theta, \qquad 1 + \cot^2 \theta = \csc^2 \theta

\sec(-\theta) = \sec \theta, \qquad \csc(-\theta) = -\csc \theta, \qquad \cot(-\theta) = -\cot \theta

\sec\left( \frac{\pi}{2} - \theta \right) = \csc \theta, \qquad \csc\left( \frac{\pi}{2} - \theta \right) = \sec \theta, \qquad \cot\left( \frac{\pi}{2} - \theta \right) = \tan \theta

Everything can be read at the point $P(\theta) = \left( \cos \theta, \sin \theta \right)$ of the lesson on trigonometric ratios: the four remaining functions were defined there as quotients of its coordinates, and the theorem on the symmetries of the sine and the cosine, proved there, gives the reflections of $P(\theta)$ in the $x$ -axis and in the line $y = x$ . The other identities are obtained by taking quotients and reciprocals. The figure uses an acute angle.

Proof

$\tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \csc \theta = \frac{1}{\sin \theta}$
The reciprocal and quotient identities are the definitions of the tangent, the cotangent, the secant and the cosecant: each is defined where its denominator does not vanish, and on that domain the product of a function and its reciprocal equals $1$ . In the figure, the sine and the cosine are the coordinates of $P(\theta)$ .
$\sin^2 \theta + \cos^2 \theta = 1 \rightarrow \frac{\resaltar{\sin^2 \theta}}{\cos^2 \theta} + 1 = \frac{1}{\resaltar{\cos^2 \theta}}, \quad 1 + \frac{\resaltar{\cos^2 \theta}}{\sin^2 \theta} = \frac{1}{\resaltar{\sin^2 \theta}}$
The point $P(\theta)$ lies at distance $1$ from the origin: this is the first Pythagorean identity, proved in the lesson on trigonometric ratios. Where $\cos \theta \neq 0$ , both of its sides are divided by $\cos^2 \theta$ , and where $\sin \theta \neq 0$ , by $\sin^2 \theta$ ; the identities of the preceding step then give the other two.
$\dato{i}{P(-\theta) = \left( \cos \theta, \resaltar{-\sin \theta} \right)}$
The reflection of $P(\theta)$ in the $x$ -axis is $P(-\theta)$ : by the theorem on the symmetries, the cosine is even and the sine is odd. In particular, $-\theta$ lies in the domain of each function exactly when $\theta$ does, because the zeros of the sine and of the cosine are symmetric.
$\sec(-\theta) = \frac{1}{\cos \theta} = \sec \theta, \quad \csc(-\theta) = \frac{1}{\resaltar{-\sin \theta}} = -\csc \theta, \quad \cot(-\theta) = \frac{\cos \theta}{\resaltar{-\sin \theta}} = -\cot \theta$
On taking reciprocals and quotients, a minus sign appears in the functions whose expression contains the sine exactly once: the secant is even, and the cosecant, the tangent and the cotangent are odd.
$\dato{c}{P\left( \frac{\pi}{2} - \theta \right) = \left( \resaltar{\sin \theta}, \resaltar{\cos \theta} \right)}$
The reflection of $P(\theta)$ in the line $y = x$ is $P\left( \frac{\pi}{2} - \theta \right)$ : by the theorem on the symmetries, this reflection interchanges the sine and the cosine.
$\sec\left( \frac{\pi}{2} - \theta \right) = \frac{1}{\resaltar{\sin \theta}} = \csc \theta, \quad \csc\left( \frac{\pi}{2} - \theta \right) = \frac{1}{\resaltar{\cos \theta}} = \sec \theta, \quad \cot\left( \frac{\pi}{2} - \theta \right) = \frac{\resaltar{\sin \theta}}{\resaltar{\cos \theta}} = \tan \theta$
On taking reciprocals and quotients, the secant and the cosecant, and the tangent and the cotangent, are interchanged as well. The domains correspond: $\frac{\pi}{2} - \theta$ lies in the domain of the secant if and only if $\theta$ lies in that of the cosecant, because $\cos\left( \frac{\pi}{2} - \theta \right) = \sin \theta$ , and the same holds for the others.

The six functions as line segments

The names of the tangent and the secant come from geometry: for an acute angle they are lengths of line segments on a tangent line and on a secant line to the unit circle. This classical figure gathers in a single drawing the six functions and the three Pythagorean identities.

TheoremThe six ratios as line segments

Let $0 < \theta < \frac{\pi}{2}$ , let $P$ be the point $P(\theta)$ of the unit circle, $Q$ the foot of its perpendicular to the $x$ -axis, $A = (1, 0)$ and $B = (0, 1)$ . The line containing the terminal side meets the vertical tangent at $A$ at a point $T$ , and the horizontal tangent at $B$ at a point $S$ . Then the cosine, the sine, the tangent, the secant, the cotangent and the cosecant of $\theta$ are, respectively, the lengths of the line segments from $O$ to $Q$ , from $Q$ to $P$ , from $A$ to $T$ , from $O$ to $T$ , from $B$ to $S$ and from $O$ to $S$ ; and the Pythagorean theorem in the triangle with vertices $O$ , $A$ and $T$ , and in the one with vertices $O$ , $B$ and $S$ , is, respectively, the second and the third Pythagorean identity.

We go through the figure one line segment at a time. The tool is the one that, in the lesson on trigonometric ratios, gave the meaning of the tangent: the line containing the terminal side passes through the origin and has slope $\tan \theta$ , so its equation is $y = x \tan \theta$ .

Proof

$P(\theta) = \left( \cos \theta, \sin \theta \right), \quad Q = \left( \cos \theta, 0 \right)$
By definition, the coordinates of $P$ are the cosine and the sine; the horizontal line segment from $O$ to $Q$ measures the abscissa, and the vertical one from $Q$ to $P$ the ordinate, both positive in the first quadrant.
$\dato{t}{T = \left( 1, \tan \theta \right)}$
The tangent to the circle at $A$ is the line $x = 1$ . Substituting $x = 1$ into the equation of the terminal side gives $T$ : the line segment from $A$ to $T$ , on the tangent line, measures $\tan \theta$ .
$T = \resaltar{\sec \theta} \cdot \left( \cos \theta, \sin \theta \right)$
The point $T$ is the point $P(\theta)$ multiplied by $\sec \theta$ , because $\sec \theta \cos \theta = 1$ . Since $P(\theta)$ lies at distance $1$ from the origin, $T$ lies at distance $\sec \theta$ : the line segment from $O$ to $T$ , on the secant line, measures the secant.
$\resaltar{1^2} + \resaltar{\tan^2 \theta} = \resaltar{\sec^2 \theta}$
The triangle with vertices $O$ , $A$ and $T$ is right-angled at $A$ , with legs $1$ and $\tan \theta$ and hypotenuse $\sec \theta$ . The Pythagorean theorem in it is the second Pythagorean identity.
$\dato{s}{S = \left( \cot \theta, 1 \right) = \csc \theta \cdot \left( \cos \theta, \sin \theta \right)}$
The tangent at $B$ is the line $y = 1$ . Substituting $y = 1$ into $y = x \tan \theta$ gives $x = \cot \theta$ ; and since $\csc \theta \sin \theta = 1$ , the point $S$ is $P(\theta)$ multiplied by $\csc \theta$ . Therefore the line segment from $B$ to $S$ measures the cotangent, and the one from $O$ to $S$ the cosecant.
$\resaltar{1^2} + \resaltar{\cot^2 \theta} = \resaltar{\csc^2 \theta}$
The triangle with vertices $O$ , $B$ and $S$ is right-angled at $B$ , with legs $1$ and $\cot \theta$ and hypotenuse $\csc \theta$ : the third Pythagorean identity follows from it. Its angle at $O$ measures $\frac{\pi}{2} - \theta$ , and in it the cotangent of $\theta$ is the opposite side: it is the tangent of the complementary angle.

The figure also shows that the secant and the cosecant of an acute angle are greater than $1$ , because they are hypotenuses of triangles with a leg of length $1$ , and that the tangent is less than the secant. The first Pythagorean identity is, likewise, the Pythagorean theorem in the triangle with vertices $O$ , $Q$ and $P$ .