Problem 36 - 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. $$\cos x\left(1+\tan ^{2} x\right)$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\cos x(1 + \tan^{2}x)\) is \(\frac{1}{\cos x}\) or \(\sec x\).

Step 1: Identify the Pythagorean Identity

In the expression, the term \(1 + \tan^{2}x\) is identified as part of the Pythagorean identity \(1 + \tan^{2}x = \sec^{2}x\).

Step 2: Substitute the Pythagorean Identity

Substitute \(\sec^{2}x\) for the term \(1 + \tan^{2}x\). The expression \(\cos x(1 + \tan^{2}x)\) now becomes \(\cos x * \sec^{2}x\).

Step 3: Solve using another identity

The identity relating \(\sec x\) and \(\cos x\) is \(\sec x = \frac{1}{\cos x}\). Therefore, \(\sec^{2}x\) can be written as \(\left(\frac{1}{\cos x}\right)^{2}\) giving the new expression as \(\cos x * \left(\frac{1}{\cos x}\right)^{2}\).

Step 4: Simplify the Expression

Simplify the expression by multiplying \(\cos x\) and \(\left(\frac{1}{\cos x}\right)^{2}\) to yield \(\frac{1}{\cos x}\).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry that relates the squares of the sine, cosine, and tangent functions.
It helps simplify expressions and equations by transforming them into equivalent forms. A classic example of a Pythagorean Identity is:

• \( \sin^2x + \cos^2x = 1 \)

This identity can be transformed by dividing the entire equation by either \(\sin^2x\) or \(\cos^2x\), leading to other forms like:

• \( 1 + \tan^2x = \sec^2x \)
• \( \cot^2x + 1 = \csc^2x \)

In problems, recognizing these identities can help you to substitute and simplify expressions. For instance, in the original exercise, \(1 + \tan^2x\) is replaced with \(\sec^2x\) using the identity \(1 + \tan^2x = \sec^2x\). Learning to quickly identify which Pythagorean Identity fits a problem will greatly enhance your trigonometric problem-solving skills.

Simplifying Expressions

Simplifying expressions is a crucial step in solving mathematical problems. In trigonometry, it often involves using identities to make expressions easier to manage.
Consider the expression \( \cos x(1 + \tan^2x) \). Initially, it seems complex, but applying the Pythagorean Identity \(1 + \tan^2x = \sec^2x\) simplifies it dramatically.
After substitution, the expression becomes \( \cos x * \sec^2x \). Further simplification involves recognizing that \( \sec x = \frac{1}{\cos x} \). Hence, \( \sec^2x = \left(\frac{1}{\cos x}\right)^2 \), and the expression simplifies to \( \cos x * \left(\frac{1}{\cos x}\right)^2 \). Calculating this multiplication results in \( \frac{1}{\cos x} \), which is a much simpler form.
Mastering the art of simplification involves:

• Identifying familiar identities and patterns in expressions.
• Systematically applying these identities to transform and reduce expressions.
• Verifying the simplified result by substituting initial values or using graphical methods.

By simplifying, you not only solve a specific problem but also increase efficiency when facing similar problems in the future.

Graphing Utility

A graphing utility is a helpful tool that can verify your simplifications of trigonometric expressions. It provides a visual or numerical confirmation of the results obtained through algebraic manipulation.
Most graphing calculators or software allow entering functions to plot their graphs or generate a table of values.
To check the simplification of \( \cos x \left(1 + \tan^2x \right) \), input both the original and simplified expressions into the graphing utility.
Generate a table of values for each expression over the same domain. The values should match, confirming the simplification is correct.
Using a graphing utility also aids in:

• Visualizing the behavior of trigonometric functions over different intervals.
• Identifying inconsistencies or errors in manual calculations.
• Reinforcing understanding through graphical representation.

Employing graphing utilities bridges the gap between theoretical math and practical application, empowering you to confidently tackle trigonometric problems.