Trigonometric Identities

Sommaire

In short: Trigonometric identities are equations that stay true for every angle where both sides are defined, and they let you rewrite one trig expression as an equivalent, simpler one.

PrecalculusTrigonometryF-TF.C.9

What is Trigonometric Identities?

The trigonometric identities are a toolbox of always-true equations. Unlike an ordinary equation, which holds only for certain angles, an identity holds for every allowed angle, so you may swap one side for the other anywhere.

The basic families are: reciprocal (\(\csc\theta = \frac{1}{\sin\theta}\), \(\sec\theta = \frac{1}{\cos\theta}\), \(\cot\theta = \frac{1}{\tan\theta}\)); quotient (\(\tan\theta = \frac{\sin\theta}{\cos\theta}\)); Pythagorean (\(\sin^2\theta + \cos^2\theta = 1\)); and cofunction (\(\sin\theta = \cos(90^{\circ} – \theta)\)).

How to spot it

A problem says verify, prove or show that two trig expressions are equal, which calls for trigonometric identities.
A messy mix of sine, cosine, tangent and secant must be reduced to a single simple term.
An equation requires one trig function to be replaced by another so that it can be solved.
A calculus-style task begins by rewriting a trig quotient in terms of sines and cosines.

The method

Pick the more complicated side and leave the other side untouched, since trigonometric identities are proven one side at a time.
Rewrite every function in terms of sine and cosine using the reciprocal and quotient identities.
Combine fractions over a common denominator and apply \(\sin^2\theta + \cos^2\theta = 1\) where it appears.
Stop when the working side matches the other, and confirm with a single test angle.

Four identity families, plus a 30° test that tan + cot equals sec · csc.

Trigonometric Identities: solved example

Problem

Verify that \(\tan\theta + \cot\theta = \sec\theta\csc\theta\).

Solution

Start with the left side and use the quotient identities: \(\frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta} = \frac{\sin^2\theta + \cos^2\theta}{\sin\theta\cos\theta}\).

The numerator equals 1, so the result is \(\frac{1}{\sin\theta\cos\theta} = \frac{1}{\cos\theta}\cdot\frac{1}{\sin\theta}\), which is \(\sec\theta\csc\theta\). At \(30^{\circ}\) both sides give 2.3094.

Try it

Question 1

Simplify \(\frac{1 – \cos^2 x}{\sin x}\).

Show the answer

The numerator equals \(\sin^2 x\), so the quotient is \(\sin x\).

Question 2

Rewrite \(\sec^2 x – 1\) as a single function.

Show the answer

From \(1 + \tan^2 x = \sec^2 x\), the expression equals \(\tan^2 x\).

Trigonometric Identities vs. Pythagorean Identity

The Pythagorean one is a single member of the larger collection, tied to the unit circle. The collection also holds reciprocal, quotient, cofunction, sum and double-angle rules. When a proof stalls, try converting everything to sine and cosine first, then look for the place where that single member applies.

Pythagorean Identity

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