Problem 76 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 73-76: Make the trigonometric substitution $$ x=a \sec \theta \text { for } 0<\theta<\pi / 2 \text { and } a>0 \text {. } $$ Simplify the resulting expression. $$ \frac{\sqrt{x^{2}-a^{2}}}{x^{2}} $$

Step-by-Step Solution

Answer

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

Step 1: Substitute x in Terms of θ

Given the trigonometric substitution \( x = a \sec \theta \), substitute this into the expression \( \frac{\sqrt{x^2 - a^2}}{x^2} \). This yields \( \frac{\sqrt{(a \sec \theta)^2 - a^2}}{(a \sec \theta)^2} \).

Step 2: Simplify the Inside of the Square Root

Calculate \( (a \sec \theta)^2 - a^2 \). This is \( a^2 \sec^2 \theta - a^2 \), which can be factored to \( a^2 (\sec^2 \theta - 1) \).

Step 3: Use Trigonometric Identity

Recall the trigonometric identity \( \sec^2 \theta - 1 = \tan^2 \theta \). Substitute this identity to get \( a^2 \tan^2 \theta \).

Step 4: Simplify the Square Root

The expression inside the square root becomes \( a^2 \tan^2 \theta \). Taking the square root gives us \( a \tan \theta \).

Step 5: Simplify the Denominator

Calculate \( (a \sec \theta)^2 \). This results in \( a^2 \sec^2 \theta \).

Step 6: Simplify the Fraction

The simplified expression is \( \frac{a \tan \theta}{a^2 \sec^2 \theta} \). Simplify this to \( \frac{\tan \theta}{a \sec^2 \theta} \).

Step 7: Expand in Terms of Sine and Cosine

Write \( \tan \theta \) and \( \sec^2 \theta \) in terms of sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \). Substitute these into the expression, resulting in \( \frac{\sin \theta}{a \cos \theta} \cdot \frac{1}{\cos^2 \theta} \).

Step 8: Final Simplification

Finally, simplify to \( \frac{\sin \theta}{a \cos^3 \theta} \). Since \( \sin \theta = \tan \theta \cdot \cos \theta \), substitute back to finish with \( \frac{\tan \theta}{a \cos^2 \theta} \).

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is crucial when dealing with trigonometric substitutions. In this exercise, the identity \( \sec^2 \theta - 1 = \tan^2 \theta \) is a key component in simplifying the expression. These relationships among trigonometric functions arise from the Pythagorean identity, which states \( \sin^2 \theta + \cos^2 \theta = 1 \). By manipulating this fundamental relationship, we derive \( \sec^2 \theta - 1 = \tan^2 \theta \), which is particularly useful in simplification tasks.

This identity aids in the process by transforming a complicated expression under the square root, making it easier to simplify. Mastery of these identities is essential not only for solving problems like this but for dealing with more complex expressions in calculus and higher mathematics.

Simplification Techniques

To simplify expressions effectively, recognize patterns and substitute identities strategically. One technique used here involves recognizing that \( (a \sec \theta)^2 - a^2 \) simplifies by factoring out \( a^2 \), resulting in \( a^2 (\sec^2 \theta - 1) \).

Upon substituting the trigonometric identity \( \sec^2 \theta - 1 = \tan^2 \theta \), the expression becomes \( a^2 \tan^2 \theta \), which is simpler and easier to work with. Further simplification of \( \sqrt{a^2 \tan^2 \theta} \) to \( a \tan \theta \) showcases how recognizing simple forms within a complex problem is crucial.

Effective simplification reduces complexity and makes the solution process more manageable, emphasizing the importance of spotting these opportunities when they arise.

Algebraic Manipulation

Algebraic manipulation involves a series of steps where expressions are rewritten to achieve a simplified form. In this exercise, after substitution and simplification inside the square root, simplifying the overall expression involves dividing \( \frac{a \tan \theta}{a^2 \sec^2 \theta} \).

This reduces to \( \frac{\tan \theta}{a \sec^2 \theta} \) by canceling out a common factor of \( a \) in the numerator and denominator. Express \( \tan \theta \) and \( \sec^2 \theta \) in terms of sine and cosine, yielding \( \frac{\sin \theta}{a \cos \theta} \cdot \frac{1}{\cos^2 \theta} \). This is further manipulated to \( \frac{\sin \theta}{a \cos^3 \theta} \), illustrating how trigonometric concepts and algebraic manipulation work hand in hand to simplify expressions.

The ability to perform algebraic manipulations skillfully can greatly enhance problem-solving skills in verifying and simplifying mathematical expressions.