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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.

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:

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



 

Videos

Simplifying Trigonometric Expressions Using Identities, Example 1
(tan3x)(csc3x)



Simplifying Trigonometric Expressions Using Identities, Example 2
sec x cos x − cos2 x
(csc2 x − 1)(sec2 x sin2 x)



 

Simplifying Trigonometric Expressions Using Identities, Example 3
(csc2 x − 1)/csc2 x
(csc2 x − cot2 x)/(tan2 x - sec2 x)

Algebraic Manipulation of Trigonometric Functions -
Part 1 Distributive Property, FOIL, Factoring.



 

Algebraic Manipulation of Trigonometric Functions -
Part 2 Simplifying Complex Fractions, Multiplying, Dividing, Adding and Subtracting Fractions.


Algebraic Manipulation of Trigonometric Functions -
Part 3 Radical Expressions: Multiplying, Dividing, Simplifying. Rationalizing the Denominator.




Algebraic Manipulation of Trigonometric Functions -
Part 4 More Examples




 

You can use the Mathway widget below to practice Trigonometry or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

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