Problem 9 - Precalculus with Calculus Previews

Question

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

Step-by-Step Solution

Answer

The expression \( \sec(-x) \cos x \) simplifies to 1.

Step 1: Apply the Even-Odd Identity to Secant

Recognize that the secant function is an even function, meaning that \( \sec(-x) = \sec(x) \). This simplifies our expression as follows:\[ \sec(-x) \cos x = \sec(x) \cos x \]

Step 2: Use the Definition of Secant

By definition, secant is the reciprocal of cosine: \( \sec(x) = \frac{1}{\cos(x)} \). Substitute this into the expression:\[ \sec(x) \cos x = \frac{1}{\cos x} \cos x \]

Step 3: Simplify the Expression

Simplifying \( \frac{1}{\cos x} \cos x \), notice that cosines in the numerator and the denominator cancel each other:\[ \frac{1}{\cos x} \cos x = 1 \]

Step 4: Final Expression

As a result of our simplification, the original expression simplifies to:\[ 1 \]

Key Concepts

Even-Odd Identities

Even-odd identities are a fundamental concept in trigonometry. They describe how functions behave under the change of signs in their arguments. In simpler terms, they help us understand what happens to trigonometric functions when we use negative angles.

Trigonometric functions are classified as either even or odd based on their symmetry. An even function satisfies the identity:

• \( f(-x) = f(x) \)

This means that the output of the function does not change, even if the angle's sign is switched from positive to negative. An example is the secant function, which is even. Thus, \( \sec(-x) = \sec(x) \).

On the other hand, an odd function satisfies:

• \( f(-x) = -f(x) \)

The sine function is an example of an odd function. It changes sign when the angle is negated, implying \( \sin(-x) = -\sin(x) \).

By recognizing whether a function is even or odd, we can simplify a lot of trigonometric expressions, as shown in this exercise.

Secant Function

The secant function, often abbreviated as sec, is one of the six fundamental trigonometric functions. It is defined in terms of the cosine function. Specifically, the secant of an angle is the reciprocal of the cosine of that angle. Put mathematically, this means:

• \( \sec(x) = \frac{1}{\cos(x)} \)

This relationship shows that the secant function can never be defined for angles where the cosine is zero, as this would require division by zero. In the case of secant, these undefined points occur every half turn, or 90 degrees plus any multiple of 180 degrees.

Understanding the secant function's reciprocal relationship with cosine is crucial when simplifying expressions using reciprocal identities. Furthermore, knowing that the secant function is even helps when dealing with negative angles, as shown when \( \sec(-x) \) simplifies to \( \sec(x) \).

Cosine Function

The cosine function is another key trigonometric function. It represents the x-coordinate of a point on the unit circle for a given angle, showing its foundation in geometry.

Cosine is an even function, meaning its value remains unchanged if the negative of an angle is taken. Mathematically, this is expressed as:

• \( \cos(-x) = \cos(x) \)

This property often comes in handy when simplifying expressions involving negative angles.

Moreover, the cosine function is critical in reciprocal identities, particularly when analysing relationships between secant and cosine. It helps build a bridge between geometric understanding and algebraic manipulation of trigonometric expressions.

Reciprocal Identities

Reciprocal identities are a powerful tool in simplifying trigonometric expressions. They relate trigonometric functions to their reciprocals, providing direct links between different functions. For instance, the secant and cosine functions are related through the reciprocal identity:

• \( \sec(x) = \frac{1}{\cos(x)} \)

These identities allow us to express one trigonometric function in terms of another, aiding in simplification. In the exercise, the reciprocal identity for secant was used to express \( \sec(x) \) as \( \frac{1}{\cos(x)} \), which then allowed the expression to be simplified to 1 by cancelling the cosine terms.

Understanding how reciprocal identities function helps solve equations and simplify complex expressions. They also provide insights into the recurring nature and symmetry of trigonometric functions across the coordinate system.