# Trigonometric Identities - Simplify Expressions

In these lessons, we will learn how to use trigonometric identities to simplify trigonometric expressions.

Related Topics: More Trigonometric Identities

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.

### Reciprocal Identities

$$\begin{array}{l}\cot \theta = \frac{1}{{\tan \theta }}\\\csc \theta = \frac{1}{{\sin \theta }}\,\,\,({\rm{some\ texts\ write\ this\ as\ }}{\mathop{\rm cosec }\nolimits} \theta )\\\sec \theta = \frac{1}{{\cos \theta }}\end{array}$$

### Quotient Identities

$$\begin{array}{l}\tan \theta = \frac{{\sin \theta }}{{\cos \theta }}\\\cot \theta = \frac{{\cos \theta }}{{\sin \theta }}\end{array}$$

### Pythagorean Identities

$$\begin{array}{l}{\sin ^2}\theta + {\cos ^2}\theta = 1\\{\tan ^2}\theta + 1 = {\sec ^2}\theta \\1 + {\cot ^2}\theta = {\csc ^2}\theta \end{array}$$

Example:

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

Solution:

$$\begin{array}{c}\frac{{\sin \theta \sec \theta }}{{{{\cos }^2}\theta }} = \frac{{\sin \theta \sec \theta }}{{\cos \theta \cos \theta }}\\ = \tan \theta \cdot \frac{{\sec \theta }}{{\cos \theta }}\\ = \tan \theta \cdot {\sec ^2}\theta \\ = \tan \theta ({\tan ^2}\theta + 1)\\ = {\tan ^3}\theta + \tan \theta \end{array}$$

Example:

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

Solution:

$$\begin{array}{c}\frac{{{{\sin }^2}\theta }}{{\cos \theta }} + \frac{{{{\cos }^2}\theta }}{{\cos \theta }} = \frac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{\cos \theta }}\\ = \frac{1}{{\cos \theta }}\\ = \sec \theta \end{array}$$

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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