Problem 43 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \( \sin^2 t \).

Step 1: Express in terms of Sine and Cosine

First, recall that the secant function is the reciprocal of the cosine function: \( \sec t = \frac{1}{\cos t} \). Substitute this into the given identity \( \frac{\sec t - \cos t}{\sec t} \).

Step 2: Simplify the Fraction

Rewrite the expression by replacing \( \sec t \) with \( \frac{1}{\cos t} \). This gives \( \frac{\frac{1}{\cos t} - \cos t}{\frac{1}{\cos t}} \).

Step 3: Simplify the Numerator

Further simplify the numerator. Starting with \( \frac{1}{\cos t} - \cos t \), factor \( \cos t \) out of the expression using a common denominator: \( \frac{1 - \cos^2 t}{\cos t} \).

Step 4: Simplify the Denominator

Simplify the denominator, which remains as \( \frac{1}{\cos t} \).

Step 5: Divide Numerator by Denominator

Use the division rule for fractions: \( \frac{a}{b} \div \frac{c}{d} = \frac{a \cdot d}{b \cdot c} \). Simplify: \( \frac{1 - \cos^2 t}{\cos t} \times \cos t = 1 - \cos^2 t \).

Step 6: Use Pythagorean Identity

Recall the identity \( \sin^2 t + \cos^2 t = 1 \). Therefore, \( \sin^2 t = 1 - \cos^2 t \).

Step 7: Verify the Identity

Thus, the left-hand side simplifies to \( \sin^2 t \), which matches the original right side of the equation. The identity \( \frac{\sec t - \cos t}{\sec t} = \sin^2 t \) is verified.

Key Concepts

Secant Function

The secant function, often denoted as \( \sec t \), is one of the six fundamental trigonometric functions. It represents the reciprocal of the cosine function. This means that it is defined as follows:

• \( \sec t = \frac{1}{\cos t} \)

Understanding the Secant Function
The function is primarily useful in problems where you are given or need the reciprocal of cosine values. It connects directly to cosine, making it vital when working with trigonometric identities and simplifications.

Role in Expressions and Equations
In trigonometry, the secant function can simplify complex expressions or identities. You can replace \( \sec t \) with its equivalent expression \( \frac{1}{\cos t} \) in equations. This is often a first step to break down or verify an identity, such as in our exercise where the secant function plays a crucial part in rewriting the original expression.

Pythagorean Identity

One of the most important identities in trigonometry is the Pythagorean identity. It is a fundamental relationship between sine and cosine, expressed as:

• \( \sin^2 t + \cos^2 t = 1 \)

Understanding the Identity
This identity allows you to express the square of the sine function in terms of the cosine function, and vice versa. In the process of solving trigonometric problems, the Pythagorean identity is often used to simplify expressions or eliminate terms.

Application in Simplification
In our exercise, after simplifying the given expression to \( 1 - \cos^2 t \), the Pythagorean identity suggests that \( \sin^2 t = 1 - \cos^2 t \). Therefore, knowing and applying the identity is crucial as it helps bridge expressions and enables verification of trigonometric identities.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a vital skill in mathematics that involves reworking trigonometric functions to simpler forms using identities, reciprocal relations, and algebraic manipulations.
Approach to Simplification
To simplify expressions like \( \frac{\sec t - \cos t}{\sec t} \), follow these steps:

• Convert all trigonometric functions to terms of sine and cosine, for example, \( \sec t = \frac{1}{\cos t} \).
• Find a common denominator for terms in the numerator or denominator as needed. This commonly involves basic algebraic manipulation.
• Replace expressions using suitable trigonometric identities like the Pythagorean identity.
• Perform arithmetic operations to simplify the final expression.

Benefits of Simplification
This process not only helps in verifying identities but also in solving equations efficiently. Simplifying expressions allows you to see relationships between trigonometric functions, making it easier to work with complex problems. In our exercise, applying these strategies led to verifying that both sides of the given identity converged to the same expression \( \sin^2 t \).