Trigonometric Identities

Section 3.1 Trigonometric Identities

In this chapter, we explore trigonometric identities and formulas, essential tools that enable us to algebraically manipulate and solve complex trigonometric equations. These identities and formulas enable us to analyze expressions in various forms, often simplifying complex expressions into ones that are easily solvable and interpretable. By doing so, we increase our ability to accurately model the world around us.

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Subsection 3.1.1 Fundamental Trigonometric Identities

An identity in mathematics is an equation that remains true for all valid values of its variables. We begin by reviewing some of the basic trigonometric identities from Chapter 1, collectively known as the fundamental trigonometric identities.

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Definition 3.1.1.

The fundamental trigonometric identities are:

Reciprocal Identities (Definition 1.4.2)

$\begin{aligned} \sin θ & = \frac{1}{\csc θ} & \cos θ & = \frac{1}{\sec θ} & \tan θ & = \frac{1}{\cot θ} \\ \csc θ & = \frac{1}{\sin θ} & \sec θ & = \frac{1}{\cos θ} & \cot θ & = \frac{1}{\tan θ} \end{aligned}$
Quotient Identities (Definition 1.4.3)

$\begin{aligned} \tan θ & = \frac{\sin θ}{\cos θ} & \cot θ & = \frac{\cos θ}{\sin θ} \end{aligned}$
Pythagorean Identities (Definition 1.5.20)

$\begin{aligned} \sin^{2} θ + \cos^{2} θ & = 1 \\ 1 + \tan^{2} θ & = \sec^{2} θ \\ 1 + \cot^{2} θ & = \csc^{2} θ \end{aligned}$
Even-Odd Identities (Definition 1.5.22)

The cosine and secant functions are even.

$\begin{aligned} \cos (- θ) & = \cos θ & \sec (- θ) & = \sec θ \end{aligned}$

The sine, cosecant, tangent, and cotangent functions are odd.

$\begin{aligned} \sin (- θ) & = - \sin θ & \csc (- θ) & = - \csc (θ) \\ \tan (- θ) & = - \tan θ & \cot (- θ) & = - \cot (θ) \end{aligned}$
Cofunction Identities (Definition 1.4.7)

$\begin{aligned} \sin θ & = \cos (\frac{π}{2} - θ), & \cos θ & = \sin (\frac{π}{2} - θ) \\ \tan θ & = \cot (\frac{π}{2} - θ), & \cot θ & = \tan (\frac{π}{2} - θ) \\ \sec θ & = \csc (\frac{π}{2} - θ), & \csc θ & = \sec (\frac{π}{2} - θ) \end{aligned}$

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Subsection 3.1.2 Simplifying Trigonometric Expressions

We use a combination of trigonometric identities, formulas, and techniques from algebra to manipulate and simplify trigonometric expressions.

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Example 3.1.2.

Simplify

\tan^{2} (x) \cdot \csc^{2} (x) .

Solution.

We can simplify this expression by writing each function in terms of sine and cosine functions:

\tan^{2} (x) \cdot \csc^{2} (x) = \frac{\sin^{2} (x)}{\cos^{2} (x)} \cdot \frac{1}{\sin^{2} (x)} = \frac{1}{\cos^{2} (x)} = \sec^{2} (x) .

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Example 3.1.3.

Simplify

\sin^{2} (x) (\cot^{2} (x) - 1) .

Solution.

We can simplify this expression by first using the Pythagorean Identity and then using the Reciprocal Identity:

\sin^{2} (x) (\cot^{2} (x) - 1) = \sin^{2} (x) (- \csc^{2} (x)) = \sin^{2} (x) (- \frac{1}{\sin^{2} (x)}) = - 1 .

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Subsection 3.1.3 Verifying Trigonometric Identities

To verify trigonometric identities, we begin with an expression on one side of the equation and manipulate it using trigonometric identities and algebraic techniques until it matches the expression on the other side.

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Remark 3.1.4. Steps for Verifying Trigonometric Identities.

To verify that an equation is an identity:

Pick an expression on one side of the equation. Often it is the more complicated expression.
Transform the expression using techniques such as trigonometric identities, rewriting in terms of sine and cosine functions, factoring, common denominator, or multiplying the numerator and denominator by the same term.
Continue manipulating until the transformed expression matches the other side of the equation.
If it is difficult to make one side equal the other, try manipulating both sides separately and make them match to reach the same result.

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Note: Unlike solving equations where we perform the same operation on both sides of the equal sign, when verifying trigonometric identities, we work with only one side and manipulate it to resemble the other side.

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