Training Trigonometry Trigonometric Identities
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Trigonometric Identities

25 min Trigonometry

Trigonometric Identities

A trigonometric identity is an equation that is true for every value of the variable for which both sides are defined. Identities are the power tools of trigonometry — they let you simplify expressions, verify equations, and solve trigonometric equations that would otherwise be intractable.

The Pythagorean identity sin²θ + cos²θ = 1 is the most fundamental, and from it you can derive two more: 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ. Together with the sum-and-difference formulas and double-angle formulas, these identities give you a complete toolkit.

This lesson covers the major identities, strategies for proving identities, and techniques for simplifying trigonometric expressions.

Interactive Explorer: Trig Identity Verifier
IdentityLHSRHS✓?
sin²θ + cos²θ = 11.00001✓
1 + tan²θ = sec²θ2.00002.0000✓
1 + cot²θ = csc²θ2.00002.0000✓
tan θ = sin θ / cos θ1.00001.0000✓

An identity is an equation that is true for all values of the variable for which both sides are defined.

Fundamental Identities

Reciprocal

$$\csc\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta}$$

Quotient

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

Pythagorean

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

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

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

Example 1

If $\sin\theta = \dfrac{3}{5}$ and $\theta$ is in QI, find $\cos\theta$ and $\tan\theta$.

$\cos^2\theta = 1 - \sin^2\theta = 1 - \frac{9}{25} = \frac{16}{25}$.

Since QI: $\cos\theta = \frac{4}{5}$.

$\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{3/5}{4/5} = \frac{3}{4}$.

Example 2

Simplify $\dfrac{\sin^2\theta}{1 - \cos\theta}$.

Replace $\sin^2\theta$ with $1 - \cos^2\theta$:

$$\frac{1 - \cos^2\theta}{1 - \cos\theta} = \frac{(1 - \cos\theta)(1 + \cos\theta)}{1 - \cos\theta} = 1 + \cos\theta$$

Example 3

Verify: $\sec^2\theta - \tan^2\theta = 1$.

This is just the Pythagorean identity $1 + \tan^2\theta = \sec^2\theta$ rearranged. ✓

Example 4

Simplify $\tan\theta \cdot \cos\theta$.

$$\frac{\sin\theta}{\cos\theta} \cdot \cos\theta = \sin\theta$$

Example 5

If $\cos\theta = -\dfrac{5}{13}$ and $\theta$ is in QIII, find $\sin\theta$.

$\sin^2\theta = 1 - \frac{25}{169} = \frac{144}{169}$.

QIII → $\sin\theta < 0$: $\sin\theta = -\frac{12}{13}$.

Practice Problems

1. If $\sin\theta = \frac{5}{13}$ (QI), find $\cos\theta$.
2. Simplify $\sin\theta \cdot \csc\theta$.
3. Simplify $1 - \sin^2\theta$.
4. If $\tan\theta = 2$ (QI), find $\sec\theta$.
5. Simplify $\dfrac{\cos\theta}{\sin\theta} \cdot \tan\theta$.
6. If $\cos\theta = \frac{7}{25}$ (QIV), find $\sin\theta$.
7. Verify: $\cot^2\theta + 1 = \csc^2\theta$.
8. Simplify $\sec\theta \cdot \cos\theta$.
9. Simplify $\dfrac{1 - \cos^2\theta}{\sin\theta}$.
10. If $\csc\theta = -3$ (QIII), find $\cos\theta$.
11. Factor $\sin^2\theta - \cos^2\theta$.
12. Simplify $\tan^2\theta - \sec^2\theta$.
Show Answer Key

1. $\frac{12}{13}$

2. $1$

3. $\cos^2\theta$

4. $\sec^2\theta = 1 + 4 = 5$; $\sec\theta = \sqrt{5}$

5. $\frac{\cos\theta}{\sin\theta} \cdot \frac{\sin\theta}{\cos\theta} = 1$

6. $\sin\theta = -\frac{24}{25}$ (QIV → sin negative)

7. Directly from Pythagorean identity ✓

8. $1$

9. $\frac{\sin^2\theta}{\sin\theta} = \sin\theta$

10. $\sin\theta = -\frac{1}{3}$; $\cos^2\theta = 1 - \frac{1}{9} = \frac{8}{9}$; QIII → $\cos\theta = -\frac{2\sqrt{2}}{3}$

11. $(\sin\theta - \cos\theta)(\sin\theta + \cos\theta)$

12. $-1$

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