Trigonometric Identities - Simplify 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.

Printable & Online Trigonometry Worksheets

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)

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How to Simplify Trigonometric Expressions Using Identities?

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

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Using Identities to Simplify Trigonometric Expressions

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

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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

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Algebraic Manipulation of Trigonometric Functions with fractions

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

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Algebraic Manipulation of Trigonometric Functions - Radical Expressions

Multiplying, Dividing, Simplifying. Rationalizing the Denominator.

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Algebraic Manipulation of Trigonometric Functions - Complex Examples

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