Problem 42 - Precalculus with Limits

Question

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer. \( \sec^2 x\left(1 - \sin^2 x\right) \)

Step-by-Step Solution

Answer

The simplified form of the given expression is 1

Step 1: Understand the given expression

The given expression is \( \sec^2 x\left(1 - \sin^2 x\right) \). This should be simplified using fundamental trigonometric identities.

Step 2: Apply Pythagorean Identity

We know the Pythagorean identity \(1 - \sin^2 x = \cos^2 x\). Substituting this into our original expression gives: \( \sec^2 x * \cos^2 x \)

Step 3: Apply Secant Identity

We also know the identity for secant, which is \( \sec x = \frac {1}{\cos x}\). Therefore, \( \sec^2 x = \frac {1}{\cos^2 x}\). Substituting this into our expression gives: \( \frac {1}{\cos^2 x} * \cos^2 x \)

Step 4: Simplification

On multiplying the expression, the \( \cos^2 x \) in the numerator and denominator cancel out, leaving us with the final simplified expression as 1

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the essential identities in trigonometry. It states that for any angle \(x\), the expression \(\sin^2 x + \cos^2 x = 1\) always holds true. This identity is derived from the Pythagorean theorem and is useful in converting between sine and cosine functions.

When you're given an expression like \(1 - \sin^2 x\), you can use the Pythagorean Identity to simplify it. Since \(1 - \sin^2 x\) is equal to \(\cos^2 x\), we substitute \(\cos^2 x\) into the expression to make it easier to work with.

This step is crucial because it transforms the problem into a form that can be simplified further using other identities. In our exercise, this meant changing \(1 - \sin^2 x\) to \(\cos^2 x\), allowing us to proceed with the simplification.

Secant Identity

The Secant Identity relates the secant function to the cosine function. It's defined as \(\sec x = \frac{1}{\cos x}\). This means that \(\sec^2 x\) is equivalent to \(\frac{1}{\cos^2 x}\).

Understanding the relationship between secant and cosine is vital in trigonometry, especially for simplifying expressions. When you see \(\sec^2 x\) in an expression, you know you can replace it with \(\frac{1}{\cos^2 x}\) to aid your simplification process.

In our example, substituting \(\sec^2 x\) with \(\frac{1}{\cos^2 x}\) is a key step. It allows the expression \(\sec^2 x \cos^2 x\) to become \(\frac{1}{\cos^2 x} \cdot \cos^2 x\), setting the stage for the simplification step where the \(\cos^2 x\) terms cancel out.

Simplification

Simplification is the process of rewriting an expression in a simpler form. This often involves cancelling terms, substituting identities, or combining like terms. The ultimate goal of simplification is to arrive at an expression that is easy to understand or further process.

In the case of our exercise, after applying both the Pythagorean and Secant identities, we were left with \(\frac{1}{\cos^2 x} \cdot \cos^2 x\). The simplification here involves recognizing that the \(\cos^2 x\) terms in the numerator and denominator cancel each other out. This leaves us with the result of 1.

Simplification is a fundamental skill in mathematics that helps reduce complex trigonometric expressions to straightforward numerical values or simpler expressions. This approach not only helps in solving homework problems but also deepens your understanding of how different trigonometric identities interact.