Basic Trigonometric Identities

Reciprocal Identities

$\csc\theta = \frac{1}{\sin\theta} \qquad \sin\theta = \frac{1}{\csc\theta}$

$\sec\theta = \frac{1}{\cos\theta} \qquad \cos\theta = \frac{1}{\sec\theta}$

$\cot\theta = \frac{1}{\tan\theta} \qquad \tan\theta = \frac{1}{\cot\theta}$

Quotient Identities

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

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

Pythagorean Identities

Primary Identity

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

Derived Identities

Divide by $\cos^2\theta$:

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

Divide by $\sin^2\theta$:

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

Alternate Forms

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

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

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

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

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

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

Even-Odd Identities

Even Functions (symmetric about y-axis)

$\cos(-\theta) = \cos\theta$

$\sec(-\theta) = \sec\theta$

Odd Functions (symmetric about origin)

$\sin(-\theta) = -\sin\theta$

$\tan(-\theta) = -\tan\theta$

$\csc(-\theta) = -\csc\theta$

$\cot(-\theta) = -\cot\theta$

Cofunction Identities

Cofunctions of complementary angles are equal.

$\sin\theta = \cos(90° - \theta) = \cos\left(\frac{\pi}{2} - \theta\right)$

$\cos\theta = \sin(90° - \theta) = \sin\left(\frac{\pi}{2} - \theta\right)$

$\tan\theta = \cot(90° - \theta) = \cot\left(\frac{\pi}{2} - \theta\right)$

$\cot\theta = \tan(90° - \theta) = \tan\left(\frac{\pi}{2} - \theta\right)$

$\sec\theta = \csc(90° - \theta) = \csc\left(\frac{\pi}{2} - \theta\right)$

$\csc\theta = \sec(90° - \theta) = \sec\left(\frac{\pi}{2} - \theta\right)$

Periodicity Identities

Period $2\pi$ (360°)

$\sin(\theta + 2\pi) = \sin\theta$

$\cos(\theta + 2\pi) = \cos\theta$

$\csc(\theta + 2\pi) = \csc\theta$

$\sec(\theta + 2\pi) = \sec\theta$

Period $\pi$ (180°)

$\tan(\theta + \pi) = \tan\theta$

$\cot(\theta + \pi) = \cot\theta$

Supplementary Angle Identities

For angles that add to $180°$ ($\pi$):

$\sin(\pi - \theta) = \sin\theta$

$\cos(\pi - \theta) = -\cos\theta$

$\tan(\pi - \theta) = -\tan\theta$

Opposite Angle Through Origin

$\sin(\pi + \theta) = -\sin\theta$

$\cos(\pi + \theta) = -\cos\theta$

$\tan(\pi + \theta) = \tan\theta$

Summary Table

Identity Type	Formula
Reciprocal	$\csc\theta = \frac{1}{\sin\theta}$
Reciprocal	$\sec\theta = \frac{1}{\cos\theta}$
Reciprocal	$\cot\theta = \frac{1}{\tan\theta}$
Quotient	$\tan\theta = \frac{\sin\theta}{\cos\theta}$
Quotient	$\cot\theta = \frac{\cos\theta}{\sin\theta}$
Pythagorean	$\sin^2\theta + \cos^2\theta = 1$
Pythagorean	$1 + \tan^2\theta = \sec^2\theta$
Pythagorean	$1 + \cot^2\theta = \csc^2\theta$
Even	$\cos(-\theta) = \cos\theta$
Odd	$\sin(-\theta) = -\sin\theta$

Using Identities to Simplify

Strategy

1. Convert everything to sine and cosine
2. Look for Pythagorean identity patterns
3. Factor when possible
4. Look for common factors to cancel

Example: Simplify $\tan\theta \cdot \cos\theta$

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

Example: Simplify $\frac{1 - \cos^2\theta}{\sin\theta}$

$\frac{1 - \cos^2\theta}{\sin\theta} = \frac{\sin^2\theta}{\sin\theta} = \sin\theta$