Problem 43 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\frac{\sec t-\cos t}{\sec t}=\sin ^{2} t$$

Step-by-Step Solution

Answer

The identity \( \frac{\sec t - \cos t}{\sec t} = \sin^2 t \) is verified.

Step 1: Understand the Identity

We need to verify that the expression \( \frac{\sec t - \cos t}{\sec t} \) is equal to \( \sin^2 t \). This means simplifying the left side to check if it matches the right side.

Step 2: Recall Trigonometric Identities

Remember that \( \sec t = \frac{1}{\cos t} \) and the Pythagorean identity \( \sin^2 t + \cos^2 t = 1 \). Using these identities will help simplify the expression.

Step 3: Rewrite Secant in Terms of Cosine

Replace \( \sec t \) with \( \frac{1}{\cos t} \): \[ \frac{\frac{1}{\cos t} - \cos t}{\frac{1}{\cos t}} \]

Step 4: Simplify the Fraction

Simplify the expression by finding a common denominator for the terms in the numerator:\[ \frac{\frac{1 - \cos^2 t}{\cos t}}{\frac{1}{\cos t}} \]

Step 5: Simplify Further Using Basic Fraction Rules

Divide the fraction in the numerator by the fraction in the denominator, which simplifies the expression:\[ 1 - \cos^2 t \]

Step 6: Use Pythagorean Identity to Simplify

Apply the identity \( 1 - \cos^2 t = \sin^2 t \). Thus, the expression simplifies to:\[ \sin^2 t \]

Step 7: Verify the Original Identity

Since the simplified expression \( \sin^2 t \) matches the original right side, the identity \( \frac{\sec t - \cos t}{\sec t} = \sin^2 t \) is verified.

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the fundamental relationships in trigonometry that connects the squares of sine and cosine functions of an angle. This identity is expressed as:

• \( \sin^2 t + \cos^2 t = 1 \)

This formula states that for any angle \( t \), the sum of the square of the sine function and the square of the cosine function is always equal to 1. It's deeply rooted in the geometry of the unit circle, where the radius is 1. On this circle, any point can be described by coordinates \((\cos t, \sin t)\), and by the definition of a circle, we get the identity. This relationship is crucial when simplifying trigonometric expressions because it allows you to switch between sine and cosine, depending on which simplification or formula you want to use. For example, when the Pythagorean identity contributed to simplify \( 1 - \cos^2 t \) into \( \sin^2 t \) in the given exercise, it allowed us to verify the original problem effectively.

Trigonometric Simplification

Trigonometric simplification involves reducing complex expressions into simpler ones using known identities and mathematical operations. In trigonometry, simplification is often a crucial step to make equations more understandable or to verify identities.

• Start by identifying known identities and replace parts of the expression.
• Look for common denominators if you're dealing with fractions.
• Simplify complex fractions by performing the division of the numerator by the denominator.

In the given exercise, the expression \( \frac{\sec t - \cos t}{\sec t} \) is simplified using these techniques. First, replace \( \sec t \) with \( \frac{1}{\cos t} \), and find a common denominator to combine the terms in the numerator. By simplifying \( \frac{1 - \cos^2 t}{\cos t} \), then dividing by \( \frac{1}{\cos t} \), we get \( 1 - \cos^2 t \). Using the Pythagorean identity, this becomes \( \sin^2 t \). Each step cleverly uses basic fraction and trigonometric properties that make simplification straightforward.

Secant and Cosine Relationship

The secant and cosine relationship is a critical component in trigonometry. Secant is the reciprocal of cosine, and is defined as:

• \( \sec t = \frac{1}{\cos t} \)

This relationship is helpful for converting expressions involving secant into ones we can more easily manipulate using cosine.
In the example problem, by transforming \( \sec t \) into \( \frac{1}{\cos t} \), we can work with fractions instead of dealing with the secant directly.
This simplification allows easier combination of terms and reveals structures like \( \frac{1-\cos^2 t}{\cos t} \) that can then be further simplified using other identities or arithmetic rules. Understanding how secant and cosine relate enables one to navigate through and simplify otherwise complicated trigonometric expressions.