Problem 37 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \frac{\tan ^{2} \beta}{1+\cos \beta}=\frac{\sec \beta-1}{\cos \beta} $$

Step-by-Step Solution

Answer

Both sides simplify to \( \frac{1 - \cos \beta}{\cos^2 \beta} \), verifying the identity.

Step 1: Understanding the Left Side

We start by examining the left side of the identity: \( \frac{\tan^2 \beta}{1+\cos \beta} \). Recall that \( \tan \beta = \frac{\sin \beta}{\cos \beta} \), therefore, \( \tan^2 \beta = \frac{\sin^2 \beta}{\cos^2 \beta} \). Substitute this into the expression to get \( \frac{\frac{\sin^2 \beta}{\cos^2 \beta}}{1+\cos \beta} = \frac{\sin^2 \beta}{\cos^2 \beta (1+\cos \beta)} \).

Step 2: Simplifying the Left Side

Now, use the Pythagorean identity \( \sin^2 \beta = 1 - \cos^2 \beta \). Substitute this in to get \( \frac{1- \cos^2 \beta}{\cos^2 \beta (1+\cos \beta)} \). Factor the numerator \( (1 - \cos^\beta)(1 + \cos \beta) \). The expression becomes \( \frac{(1 - \cos \beta)(1 + \cos \beta)}{\cos^2 \beta (1+\cos \beta)} \). Cancel \( (1 + \cos \beta) \) from the numerator and the denominator, simplifying the left side to \( \frac{1 - \cos \beta}{\cos^2 \beta} \).

Step 3: Working on the Right Side

Now, let's simplify the right side of the identity: \( \frac{\sec \beta - 1}{\cos \beta} \). Recall that \( \sec \beta = \frac{1}{\cos \beta} \), so substituting gives us \( \frac{\frac{1}{\cos \beta} - 1}{\cos \beta} = \frac{1 - \cos \beta}{\cos \beta^2} \).

Step 4: Comparison of Both Sides

Now, both sides of the identity are \( \frac{1 - \cos \beta}{\cos^2 \beta} \). This confirms that the given identity is indeed correct.

Key Concepts

Trigonometric Functions

Trigonometric functions are foundational in mathematics, especially in the study of angles and their relationships. These functions include sine (\( \sin \)), cosine (\( \cos \)), tangent (\( \tan \)), and their reciprocals cosecant (\( \csc \)), secant (\( \sec \)), and cotangent (\( \cot \)). Each of these functions relates to a right triangle and the circle's unit properties, making them crucial in geometry, physics, and engineering.

• The sine function (\( \sin \)) represents the opposite side over the hypotenuse in a right triangle.
• The cosine function (\( \cos \)) represents the adjacent side over the hypotenuse.
• The tangent function (\( \tan \)) is the ratio of the sine and cosine functions, or opposite over adjacent.

In the given identity problem, the tangent and secant functions are particularly important. Understanding these relationships, such as \( \tan \beta = \frac{\sin \beta}{\cos \beta} \) and \( \sec \beta = \frac{1}{\cos \beta} \), is essential to verifying and manipulating trigonometric identities.

Pythagorean Identity

The Pythagorean identity is one of the fundamental identities in trigonometry. It comes from the Pythagorean theorem and relates the squares of the sine and cosine functions to 1. The identity is written as:\[\sin^2 \theta + \cos^2 \theta = 1\]

This identity can be rearranged to find other useful relations:

• \( \sin^2 \theta = 1 - \cos^2 \theta \)
• \( \cos^2 \theta = 1 - \sin^2 \theta \)

The Pythagorean identity is crucial in simplifying and proving trigonometric statements. In the original exercise, it transformed \( \tan^2 \beta \), which was expressed as \( \frac{\sin^2 \beta}{\cos^2 \beta} \), into a simpler form suitable for comparison and simplification. Understanding this identity allows mathematicians to manipulate complex expressions into simpler, equivalent ones.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves using algebraic manipulation to convert complex trigonometric expressions into simpler forms. This step is crucial in verifying identities or solving equations. The goal is always to reduce complex functions into basic trigonometric functions if possible.

When simplifying, it is useful to:

• Use known identities like the Pythagorean identities, reciprocal identities, and angle sum formulas.
• Factor expressions wherever applicable.
• Cancel common terms in numerators and denominators.

In the exercise, the simplification process involved factoring and canceling terms on both sides of the identity. Initially, the left side was expressed as \( \frac{\tan^2 \beta}{1+\cos \beta} = \frac{\sin^2 \beta}{\cos^2 \beta (1+\cos \beta)} \). By applying the Pythagorean identity and canceling like terms, it reduced to match the right side of the equation. Such simplification is key to verifying the identity efficiently and requires an understanding of how different trigonometric expressions interact with each other.