Trigonometric Identities and Formulas: A Precalculus Study Guide

Proving Identities

Definition and Examples of Identities

An identity is a mathematical statement that is always true for all values of the variables involved. In algebra and trigonometry, identities are used to simplify expressions and solve equations.

• Difference of Squares: $(a + b)(a - b) = a^2 - b^2$

• Square of a Binomial: $(a + b)^2 = a^2 + 2ab + b^2$

• Sum of Reciprocals: $\frac{1}{a} + \frac{1}{b} = \frac{b + a}{ab}$, where $a \neq 0, b \neq 0$

Pythagorean Identities

Derivation from the Pythagorean Theorem

The Pythagorean Theorem relates the sides of a right triangle:

• $(\text{opp})^2 + (\text{adj})^2 = (\text{hyp})^2$

Dividing both sides by $(\text{hyp})^2$ gives the first Pythagorean identity:

• $\sin^2 \theta + \cos^2 \theta = 1$

Other Pythagorean Identities

• Dividing $\sin^2 \theta + \cos^2 \theta = 1$ by $\cos^2 \theta$:

• $\tan^2 \theta + 1 = \sec^2 \theta$

• Dividing by $\sin^2 \theta$ (Additional info: this yields another identity):

• $1 + \cot^2 \theta = \csc^2 \theta$

Identifying Non-Identities

Testing for Identities

Not all equations are identities. To show an equation is not an identity, find a value where the equality fails.

• Example: $\sin \theta = \sqrt{1 - \cos^2 \theta}$ is not an identity because $\sin \theta$ can be negative, but the right side is always non-negative.

• Example: $\csc \theta = \sqrt{1 - \cot^2 \theta}$ is not an identity for similar reasons.

Proving Trigonometric Identities

Sample Proofs

• $(\sin \theta - \cos \theta)(\sin \theta + \cos \theta) = 1 - 2\cos^2 \theta$

• $\frac{1}{1 - \cos \theta} + \frac{1}{1 + \cos \theta} = 2 \csc^2 \theta$

• $\cot^2 \theta + \tan^2 \theta = \sec^2 \theta + \csc^2 \theta - 2$

• $\frac{\cos \theta}{1 - \sin \theta} = \sec \theta + \tan \theta$

To prove an identity, manipulate one or both sides using algebraic and trigonometric properties until both sides are equal.

Trigonometric Substitutions

Common Substitutions Based on Pythagorean Identities

Trigonometric substitutions are useful in calculus and integration, but are based on precalculus identities:

Expression	Substitution	Resulting Identity
$y = \sqrt{1 - x^2}$	$x = \sin \theta$	$y = \cos \theta$
$y = \sqrt{x^2 + 1}$	$x = \tan \theta$	$y = \sec \theta$
$y = \sqrt{x^2 - 1}$	$x = \sec \theta$	$y = \tan \theta$

Example: If $y = \sqrt{x^2 - 4}$, use $x = 2 \sec \theta$ to rewrite $y$ in terms of $\theta$.

Sum and Difference Formulas

Formulas for Sine, Cosine, and Tangent

• $\cos(\theta_1 \pm \theta_2) = \cos \theta_1 \cos \theta_2 \mp \sin \theta_1 \sin \theta_2$

• $\sin(\theta_1 \pm \theta_2) = \sin \theta_1 \cos \theta_2 \pm \sin \theta_2 \cos \theta_1$

• $\tan(\theta_1 \pm \theta_2) = \frac{\tan \theta_1 \pm \tan \theta_2}{1 \mp \tan \theta_1 \tan \theta_2}$

These formulas allow the calculation of trigonometric values for sums and differences of angles.

Applications and Examples

• Find $\sin(15^\circ)$ using $15^\circ = 45^\circ - 30^\circ$: Use the sine difference formula.

• Find $\cos(\frac{\pi}{12})$ using $\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$: Use the cosine difference formula.

• Verify the identity $\sin x = \cos(x - \frac{\pi}{2})$ both geometrically and algebraically.

• Find the value: $\cos(57^\circ)\cos(33^\circ) - \sin(57^\circ)\sin(33^\circ)$ (use the cosine sum formula).

• Find $\sin(u + v)$ given $\tan u = \frac{3}{4}$, $u$ in quadrant I, $\sin v = \frac{5}{13}$, $v$ in quadrant II.

• Find $\cos(u + v)$ given $\sin u = -\frac{4}{5}$, $u$ in quadrant IV, $\sin v = \frac{12}{13}$, $v$ in quadrant II.

• Find $\cos(\sin^{-1}(-3/5) + \cos^{-1}(-2/3))$.

• Find $\sin(\tan^{-1}(-3/4) + \cos^{-1}(-4/5))$.

Double Angle Formulas

Formulas for Sine, Cosine, and Tangent

• $\sin 2\theta = 2 \sin \theta \cos \theta$

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

• $\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}$

Alternative Forms for Cosine Double Angle

• $\cos 2\theta = 1 - 2\sin^2 \theta$

• $\cos 2\theta = 2\cos^2 \theta - 1$

Examples

• If $\sec \theta = -\sqrt{3}$ and $\sin \theta > 0$, find $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$.

• If $\cos \theta = -\frac{12}{13}$ and $\tan \theta < 0$, find $\sin 2\theta$, $\cos 2\theta$, and $\tan 2\theta$.

Summary Table: Key Trigonometric Identities

Identity Name	Formula
Pythagorean	$\sin^2 \theta + \cos^2 \theta = 1$
Pythagorean (tan/sec)	$\tan^2 \theta + 1 = \sec^2 \theta$
Pythagorean (cot/csc)	$1 + \cot^2 \theta = \csc^2 \theta$
Sum Formula (Sine)	$\sin(a + b) = \sin a \cos b + \cos a \sin b$
Sum Formula (Cosine)	$\cos(a + b) = \cos a \cos b - \sin a \sin b$
Double Angle (Sine)	$\sin 2a = 2 \sin a \cos a$
Double Angle (Cosine)	$\cos 2a = \cos^2 a - \sin^2 a$

Additional info: Some context and examples have been expanded for clarity and completeness, as is standard in a mini-textbook study guide.