Simplifying and Proving Trig Identities
A long trig expression like (sec x - cos x) / tan x looks like it must equal something messy. Convert each piece to sine and cosine, clean it up, and it collapses to plain sin x. Proving an identity is the same skill turned into an argument: show that one side, rewritten step by step, becomes the other side.
By the end of this page you can:
- Rewrite a trig expression using reciprocal, Pythagorean, and even-odd identities until it is in simplest form.
- Prove a given identity by transforming one side into the other.
- Choose a productive first move: convert to sine and cosine, or combine fractions over a common denominator.
The floor. An identity is true for every value where both sides are defined. To prove one, work on a single side and rewrite it with known identities until it matches the other side. Never assume the statement is true and operate on both sides at once.
Quick Reference
| Field | Value |
|---|---|
| Textbook | Stewart, Redlin, Watson 7e |
| Chapter | 7 |
| Section | 7.1 Trigonometric Identities |
| Open alternate | OpenStax Precalculus 2e, 7.1 Solving Trigonometric Equations with Identities |
| Difficulty | Core |
| Time | About 30 minutes |
Before You Start
The Idea
Most proofs reduce to a small toolkit. Convert every function to sine and cosine, combine fractions over a common denominator, factor when a quadratic-looking pattern appears, and apply the Pythagorean identity to trade one squared function for another. The guiding rule of method is to start from the more complicated side and simplify it toward the simpler side, because there is more room to make productive cuts.
The single most-used tool is the Pythagorean identity. It lets you replace 1 - cos^2 x with sin^2 x, or sec^2 x - 1 with tan^2 x, which is often the move that makes a fraction cancel.
Key Formulas
The Pythagorean identity and its rearrangements:
\[ \sin^2 x + \cos^2 x = 1, \qquad 1 - \cos^2 x = \sin^2 x, \qquad 1 - \sin^2 x = \cos^2 x \]
The reciprocal and quotient identities used to convert to sine and cosine:
\[ \tan x = \frac{\sin x}{\cos x}, \qquad \sec x = \frac{1}{\cos x}, \qquad \csc x = \frac{1}{\sin x}, \qquad \cot x = \frac{\cos x}{\sin x} \]
Source: Stewart 7e, Section 7.1 Trigonometric Identities.
Worked Example
Prove the identity:
\[ \frac{1 - \cos^2 x}{\sin x} = \sin x \]
Work on the left side alone and rewrite it until it equals the right side.
Show the proof
Start with the left side and apply the rearranged Pythagorean identity to the numerator.
\[ \frac{1 - \cos^2 x}{\sin x} = \frac{\sin^2 x}{\sin x} \]
Then cancel one factor of sin x, which is valid wherever sin x is not zero (exactly where the original left side is defined).
\[ \frac{\sin^2 x}{\sin x} = \sin x \]
The left side now equals the right side, so the identity is proved.
The cancellation step is the place to be careful: you may divide by sin x only because the expression already requires sin x to be nonzero. Dividing by a factor that could be zero is the most common way a proof goes wrong.
Connections
This builds on the fundamental identities, since the reciprocal, Pythagorean, and even-odd identities are the only tools a proof rewrites with. It also leans on signs and domains of the trig functions, which tells you where a step like cancellation stays valid. From here the work leads into the addition and subtraction formulas, which introduce new identities to prove and simplify with, and into solving basic trigonometric equations, where the same simplifying moves turn a hard equation into one you can solve.
Resources
- Primary: Stewart, Redlin, Watson 7e, Section 7.1 Trigonometric Identities.
- Alternate (open): OpenStax Precalculus 2e, Section 7.1 Solving Trigonometric Equations with Identities.
Last updated: 2026-06-24