Related Topics: More Trigonometric Identities

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 \(\frac{{\sin \theta \sec \theta }}{{{{\cos }^2}\theta }}\)

**Simplifying Trigonometric Expressions Using Identities**

Example:

(tan^{3}*x*)(csc^{3}*x*)
**How to Simplify Trigonometric Expressions Using Identities?**

Example:

sec* x* cos* x* − cos^{2} *x*

(csc^{2} *x* − 1)(sec^{2} *x* sin^{2} *x*)
**Using Identities to Simplify Trigonometric Expressions**

Example:

(csc^{2} *x* − 1)/csc^{2} *x*

(csc^{2} *x* − cot^{2} *x*)/(tan^{2} *x* - sec^{2} *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)

sin^{2}x cos^{2}x + cos^{4}x

**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**

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.

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.

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:**

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

Example:

(tan

Example:

sec

(csc

Example:

(csc

(csc

Distributive Property, FOIL, Factoring.

Examples:

cos y(tan y - sec y)

(sin x + cos x)(sin x - cos x)

sin

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

Multiplying, Dividing, Simplifying. Rationalizing the Denominator.

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

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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