# Trigonometric Identities - Simplify Expressions

In these lessons, we will learn to use trigonometric identities to simplify trigonometric expressions. These video lessons with examples, step-by-step solutions, and explanations help High School Algebra 2 students learn to use trigonometric identities to simplify trigonometric expressions.

The following are some common trigonometric identities: Reciprocal Identities, Quotient Identities and Pythagorean Identities. Scroll down the page for examples and solutions using the identities to simply trigonometric expressions.

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

Solution:
$$\frac{{\sin \theta \sec \theta }}{{{{\cos }^2}\theta }} = \frac{{\sin \theta \sec \theta }}{{\cos \theta \cos \theta }}$$
$$= \tan \theta \cdot \frac{{\sec \theta }}{{\cos \theta }}$$
= tan θ • sec2 θ
= tan θ (tan2 θ + 1)
= tan3 θ + tan θ

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

Solution:
$$\frac{{{{\sin }^2}\theta }}{{\cos \theta }} + \frac{{{{\cos }^2}\theta }}{{\cos \theta }}$$
$$= \frac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{\cos \theta }}$$
$$= \frac{1}{{\cos \theta }}$$
= sec θ

Simplifying Trigonometric Expressions Using Identities

Example:
(tan3x)(csc3x)

How to Simplify Trigonometric Expressions Using Identities?

Example:
sec x cos x − cos2 x
(csc2 x − 1)(sec2 x sin2 x)

Using Identities to Simplify Trigonometric Expressions

Example:
(csc2 x − 1)/csc2 x
(csc2 x − cot2 x)/(tan2 x - sec2 x)

Algebraic Manipulation of Trigonometric Functions

Distributive Property, FOIL, Factoring.

Example:
cos y(tan y - sec y)
(sin x + cos x)(sin x - cos x)
sin2x cos2x + cos4x

Algebraic Manipulation of Trigonometric Functions with fractions

Simplifying Complex Fractions, Multiplying, Dividing, Adding and Subtracting Fractions.

Algebraic Manipulation of Trigonometric Functions - Radical Expressions

Multiplying, Dividing, Simplifying. Rationalizing the Denominator.

Algebraic Manipulation of Trigonometric Functions - Complex Examples

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.