Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. $$\frac{1+\tan ^{2} \theta}{\sec ^{2} \theta}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is 1.

Step 1: Recognize the identity \(\tan^2\theta = \sec^2\theta - 1\)

Substitute \(\tan^2\theta\) with \(\sec^2\theta - 1\) in the given expression. Thus, the new expression becomes \(\frac{1+(\sec^2\theta - 1)}{\sec^2\theta}\).

Step 2: Simplify the expression

Simplify the expression to get \(\frac{\sec^2\theta}{\sec^2\theta}\) which simplifies to 1.

Step 3: Check your result

Use a graphing utility to confirm that the original and simplified expressions yield the same result for various values of \(\theta\).

Key Concepts

Trigonometric Identities

When working with trigonometry, certain equations called trigonometric identities are pivotal for simplifying expressions and solving problems. These identities are equalities that involve trigonometric functions and are true for all values of the variables within their domain. The most widely used are the Pythagorean identities, quotient identities, and reciprocal identities.

For instance, the Pythagorean identity \( \tan^2 \theta + 1 = \text{sec}^2 \theta \) is an indispensable tool. It emerges from the more familiar Pythagorean theorem applied to the unit circle where the radius is 1. This trigonometric identity relates the square of the tangent function to the square of the secant function, which helps simplify complex expressions into simpler forms.

Simplifying Trigonometric Expressions

Simplification of trigonometric expressions is a methodical process, vital for solving trigonometry problems efficiently. To start, familiarize yourself with basic trigonometric identities. Identify parts of the expression that resemble these identities, and then systematically apply them to simplify the expression step by step.

Consider the exercise \( \frac{1 + \tan^2 \theta}{\text{sec}^2 \theta} \). By recognizing that \( \tan^2 \theta \) is part of the Pythagorean identity, one can replace it to reveal a simpler structure within the complex expression. This step is crucial as it transforms the original problem into a more manageable form. In the end, such techniques lead to a significant reduction in the complexity of trigonometric expressions, making them easier to understand and solve.

Graphing Utility for Trigonometry

A graphing utility is a powerful aid that can confirm the correctness of algebraic manipulations involving trigonometric functions. After simplifying an expression, it's wise to use such a tool to visually inspect and numerically verify the equivalence of the original and simplified expressions over a range of angles.

For the given exercise, plotting \( \frac{1 + \tan^2 \theta}{\text{sec}^2 \theta} \) and the simplified version, which is 1, will show overlapping graphs, signifying their equivalence. This visual check acts as a practical affirmation of the mathematical steps taken during simplification. Accurate graphs, generated from the utility, will agree at every selected value of \( \theta \) and help boost confidence in the understanding and application of trigonometric identities in simplification processes.