Trigonometric Identities - Simplify Expressions

Trigonometric identities are equalities that involve trigonometric functions and are true for every value of the variables for which both sides of the equality are defined. They are fundamental in simplifying trigonometric expressions, solving trigonometric equations, and proving other 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.

 

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1. Reciprocal Identities
Sine and Cosecant:
The cosecant (csc) function is the reciprocal of the sine (sin) function.
\(\csc \theta = \frac{1}{\sin \theta}\)
or
\(\sin \theta = \frac{1}{\csc \theta}\)
Cosine and Secant:
The secant (sec) function is the reciprocal of the cosine (cos) function.
\(\cos \theta = \frac{1}{\sec \theta}\)
or
\(\sec \theta = \frac{1}{\cos \theta}\)
Tangent and Cotangent:
The cotangent (cot) function is the reciprocal of the tangent (tan) function
\(\tan \theta = \frac{1}{\cot \theta}\)
or
\(\cot \theta = \frac{1}{\tan \theta}\)

2. Quotient Identities
These identities express the tangent and cotangent functions in terms of sine and cosine functions.
\(\tan \theta = \frac{\sin \theta }{\cos \theta}\)

\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)

3. Pythagorean Identities
Derived from the Pythagorean theorem, these identities relate the squares of the primary trigonometric functions.
\(\sin^2 \theta + \sin^2 \theta = 1\)
\(1 + \tan^2 \theta = \sec^2 \theta\)
\(1 + \cot^2 \theta = \csc^2 \theta\)

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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