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


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