Problem 71 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 69-72: Make the trigonometric substitution $$ x=a \tan \theta \text { for }-\pi / 2<\theta<\pi / 2 \text { and } a>0 \text {. } $$ Simplify the resulting expression. $$ \frac{1}{x^{2}+a^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{\cos^2 \theta}{a^2}\).

Step 1: Identify the Trigonometric Substitution

For this problem, we are given that the substitution is \( x = a \tan \theta \). This means we will replace every occurrence of \( x \) in the expression with \( a \tan \theta \).

Step 2: Substitute \( x \) in the Expression

Substitute \( x = a \tan \theta \) in the given expression \( \frac{1}{x^2 + a^2} \). This gives us: \[\frac{1}{(a \tan \theta)^2 + a^2} \]

Step 3: Simplify the Expression Inside the Denominator

Calculate \((a \tan \theta)^2\). This equals to \(a^2 \tan^2 \theta \). So the expression becomes:\[\frac{1}{a^2 \tan^2 \theta + a^2}\]

Step 4: Factor Out \(a^2\) from the Denominator

Factor \(a^2\) out of the terms in the denominator:\[\frac{1}{a^2(\tan^2 \theta + 1)}\]

Step 5: Use the Pythagorean Identity

Recall the identity \(\tan^2 \theta + 1 = \sec^2 \theta\). Replace \(\tan^2 \theta + 1\) with \(\sec^2 \theta\):\[\frac{1}{a^2 \sec^2 \theta}\]

Step 6: Simplify Further

Since \( \sec \theta = \frac{1}{\cos \theta} \), we can rewrite \( \sec^2 \theta \) as \( \frac{1}{\cos^2 \theta} \). Simplify the expression:\[\frac{1}{a^2} \cdot \cos^2 \theta = \frac{\cos^2 \theta}{a^2}\]

Step 7: Final Simplified Expression

The final simplified expression after trigonometric substitution and simplification is:\[\frac{\cos^2 \theta}{a^2}\]

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in simplifying complex mathematical expressions, especially those involving trigonometric functions like sine, cosine, and tangent. In the context of our problem, the key identity used is the Pythagorean identity: \( \tan^2 \theta + 1 = \sec^2 \theta \). This identity allows us to replace a combination of trigonometric functions with another equivalent trigonometric function, making it easier to handle and simplify expressions.

Using trigonometric identities requires familiarity with a few key principles:

• The Pythagorean identities – these relate the squares of the basic trigonometric functions to each other, like \( \sin^2 \theta + \cos^2 \theta = 1 \) and \( \tan^2 \theta + 1 = \sec^2 \theta \).
• The ratio identities – such as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \).
• The reciprocal identities – like \( \csc \theta = \frac{1}{\sin \theta} \) and \( \cot \theta = \frac{1}{\tan \theta} \).

Mastering these identities allows for more elegant solutions to trigonometric problems through substitution and transformation of expressions.

Simplification

Simplification is crucial in solving trigonometric expressions, as it allows us to express complex problems in a more manageable form. In this exercise, simplification occurs primarily through substitution and algebraic manipulation.

Here's how simplification works step-by-step in our problem:

• First, we substitute \( x = a \tan \theta \) into the expression \( \frac{1}{x^2 + a^2} \). This changes the expression to \( \frac{1}{a^2 \tan^2 \theta + a^2} \).
• Next, factor \( a^2 \) out of the terms in the denominator, simplifying it to \( \frac{1}{a^2(\tan^2 \theta + 1)} \).
• Finally, using the Pythagorean identity, replace \( \tan^2 \theta + 1 \) with \( \sec^2 \theta \), simplifying the expression to \( \frac{1}{a^2 \sec^2 \theta} \).

These simplification steps lead to a much more straightforward expression, \( \frac{\cos^2 \theta}{a^2} \), by canceling and rearranging terms to reach the final result.

Pythagorean Identity

The Pythagorean identity is a fundamental principle in trigonometry, often used to simplify expressions involving squared trigonometric terms. This identity states that for a given angle \( \theta \), \( \tan^2 \theta + 1 = \sec^2 \theta \).

In our specific problem, this identity is the key to transforming the expression \( \tan^2 \theta + 1 \), appearing in the denominator, into \( \sec^2 \theta \). This transformation via the Pythagorean identity enables further simplification:

• The expression transitions from \( \frac{1}{a^2(\tan^2 \theta + 1)} \) to \( \frac{1}{a^2 \sec^2 \theta} \).
• Recognizing \( \sec^2 \theta \) as \( \frac{1}{\cos^2 \theta} \) allows the expression to simplify further into \( \frac{\cos^2 \theta}{a^2} \).

This showcases the power of the Pythagorean identity in simplifying complex trigonometric expressions and highlights the importance of understanding and applying fundamental trigonometric relationships.