Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to check your result graphically. $$\sec ^{2} x \tan ^{2} x+\sec ^{2} x$$

Step-by-Step Solution

Answer

The simplified form of \( \sec^{2}x \tan^{2}x+\sec^{2}x \) is \( 2 \tan^{2}x + \tan^{4}x + 1 \)

Step 1: Identify the formula

In simplifying trigonometric expressions, it's essential to identify existing identities. In this case, we recognize that \( \sec^{2}x = 1 + tan^{2}x \)

Step 2: Apply the identity

Substitute \( \sec^{2}x = 1 + tan^{2}x \) into the given expression to reduce its complexity. By substituting we get: \( (1 + \tan^{2}x) \tan^{2}x + (1+ \tan^{2}x) = \tan^{2}x + \tan^{4}x + 1+ \tan^{2}x \)

Step 3: Simplify the result

Combine like terms: \( 2 \tan^{2}x + \tan^{4}x + 1 \)

Step 4: Checking the Result Graphically

To check this result graphically, you'd plot the original expression \( \sec^{2}x \tan^{2}x+\sec^{2}x \) and the simplified expression \( 2 \tan^{2}x + \tan^{4}x + 1 \) on a graphing calculator. These two curves should overlap because they are equivalent expressions, hence confirming the algebraic simplification.

Key Concepts

Trigonometric Simplification

Trigonometric simplification involves using known trigonometric identities to rewrite complex expressions in a simpler form. This helps in both solving equations and understanding relationships between different trigonometric functions. In the given exercise, we focus on simplifying the expression \(\sec^{2}x \tan^{2}x+\sec^{2}x\).

To simplify trigonometric expressions, you often use identities like the Pythagorean identity: \(\sec^{2}x = 1 + \tan^{2}x\). Applying this identity allows us to replace \(\sec^{2}x\) in the original expression, reducing it to a form that features fewer distinct trigonometric functions.

Once identities are applied, simplifying further involves combining like terms, just as you would in algebraic expressions. In this instance, after substitution, the expression becomes \(2 \tan^{2}x + \tan^{4}x + 1\). This process of simplification not only makes calculations easier but also forms a foundation for solving more complex trigonometric problems.

Graphing Utility

A graphing utility, such as a graphing calculator or software like Desmos, is an invaluable tool for visualizing mathematical expressions. It allows you to plot functions and visually compare their behavior.

By graphing the original and simplified expressions, \(\sec^{2}x \tan^{2}x+\sec^{2}x\) and \(2 \tan^{2}x + \tan^{4}x + 1\), you can confirm their equivalence. When plotted, these expressions should overlap completely, indicating that the algebraic simplification is correct.

Using graphing utilities is especially helpful when learning trigonometry because it allows you to see the immediate impact of algebraic changes. This visual approach can make understanding abstract concepts more concrete.

Secant and Tangent Functions

Secant and tangent are fundamental trigonometric functions that relate to the ratios found in right triangles. The secant function, \(\sec x\), is the reciprocal of the cosine function. Mathematically, it is defined as \(\sec x = \frac{1}{\cos x}\).

The tangent function, \(\tan x\), represents the ratio of the sine and cosine functions: \(\tan x = \frac{\sin x}{\cos x}\). These functions often appear together due to their interrelated nature through trigonometric identities like \(\sec^{2}x = 1 + \tan^{2}x\).

Understanding these functions and their identities is vital for both simplifying expressions and solving equations in trigonometry. Recognizing how they transform through different identities helps provide deeper insights into their behavior and interactions.