Problem 1 - Precalculus with Calculus Previews

Question

Use the fundamental identities and the even-odd identities to simplify each expression. $$ \sec t \cos t $$

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Identify Identities Involved

Our goal is to simplify the expression \( \sec t \cos t \). The identity for secant is \( \sec t = \frac{1}{\cos t} \). We'll use this identity to simplify the given expression.

Step 2: Substitute Secant Identity

Replace \( \sec t \) in the expression with its identity \( \frac{1}{\cos t} \). So, the expression becomes \( \frac{1}{\cos t} \cos t \).

Step 3: Simplify the Expression

Notice that \( \frac{1}{\cos t} \cos t \) is equivalent to \( \cos t \times \frac{1}{\cos t} \). This simplifies to 1 since \( \cos t \) and \( \frac{1}{\cos t} \) are multiplicative inverses.

Key Concepts

Fundamental Identities

Fundamental identities are the backbone of simplifying trigonometric expressions. They serve as powerful tools that allow us to manipulate and transform complex trigonometric expressions into simpler forms.
The fundamental identities include the following:

• The reciprocal identities:
• \( \sin t = \frac{1}{\csc t} \)
• \( \cos t = \frac{1}{\sec t} \)
• \( \tan t = \frac{1}{\cot t} \)
• The quotient identities:
• \( \tan t = \frac{\sin t}{\cos t} \)
• \( \cot t = \frac{\cos t}{\sin t} \)
• The Pythagorean identities:
• \( \sin^2 t + \cos^2 t = 1 \)
• \( 1 + \tan^2 t = \sec^2 t \)
• \( 1 + \cot^2 t = \csc^2 t \)

Understanding these identities is crucial as they often offer the simplest path to simplify trigonometric expressions. In our given exercise, we utilized the reciprocal identity \(\sec t = \frac{1}{\cos t}\) to transform and simplify the expression \(\sec t \cos t\). This plays a key role in resolving the complexity inherent in trigonometric expressions.

Even-Odd Identities

Even-odd identities help us understand the symmetry of trigonometric functions. These identities are invaluable when simplifying expressions or solving equations because they tell us how functions behave when their input sign changes.
Even functions are symmetric with respect to the y-axis, while odd functions are symmetric with respect to the origin. Here’s a quick breakdown:

• Even Function: \( \cos(-t) = \cos t \)
• Odd Functions: \( \sin(-t) = -\sin t \)
• \( \tan(-t) = -\tan t \)
• \( \csc(-t) = -\csc t \)
• \( \sec(-t) = \sec t \)
• \( \cot(-t) = -\cot t \)

For our problem \(\sec t \cos t\), while even-odd identities weren't directly necessary, having an understanding of how they work is crucial. It ensures we can manage expressions correctly, especially when negative angles or reflections are present. Knowing when a function keeps its sign or changes can make all the difference in complex simplifications.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves reducing them to their simplest form, which often requires using identities. This process can make equations easier to solve and computations less cumbersome.
To simplify expressions like \(\sec t \cos t\), follow these steps:

• Identify the identities that apply: Look for opportunities to use the fundamental identities, like reciprocal and Pythagorean identities.
• Substitute: Replace trigonometric functions with their equivalent identities, as done with \(\sec t\). Here, substituting \(\sec t\) with \(\frac{1}{\cos t}\) converted the expression to \(\frac{1}{\cos t}\times \cos t\).
• Simplify: Use basic arithmetic and algebraic principles. In this exercise, the \(\cos t\) terms canceled each other out, simplifying to 1, because any number divided by itself equals 1 (assuming it's not zero).

This simplification process is a critical skill in mathematics, providing clarity and efficiency. By mastering how to simplify, you'll unlock the ability to tackle more complex trigonometric problems with confidence and ease.