 # Trigonometric Identities - Simplify Expressions

Related Topics:
Trigonometric Identities
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Examples, solutions, videos, and lessons to help High School Algebra 2 students learn to use trigonometric identities to simplify trigonometric expressions. In these lessons, we will learn how 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.
Examples:
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

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