Problem 16 - Precalculus Mathematics for Calculus

Question

Simplify the trigonometric expression. $$\frac{\tan x}{\sec (-x)}$$

Step-by-Step Solution

Answer

The expression simplifies to \( \sin x \).

Step 1: Simplify the denominator

The expression given is \( \frac{\tan x}{\sec (-x)} \). Recall that the secant function is even, meaning that \( \sec(-x) = \sec x \). So, we can rewrite the expression as \( \frac{\tan x}{\sec x} \).

Step 2: Use trigonometric identities

Recall the trigonometric identities: \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \). Replace these identities in the expression: \[\frac{\tan x}{\sec x} = \frac{\frac{\sin x}{\cos x}}{\frac{1}{\cos x}}.\]

Step 3: Simplify the fraction

Simplify the compound fraction by multiplying the numerator by the reciprocal of the denominator. The expression becomes: \[\frac{\sin x}{\cos x} \times \cos x = \sin x.\]The \( \cos x \) terms cancel each other out.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in simplifying expressions involving trigonometric functions. They allow us to rewrite functions using known equivalences and properties. A couple of key identities you need to remember for simplifications are:

• The tangent identity: \[ \tan x = \frac{\sin x}{\cos x} \]
• The secant identity: \[ \sec x = \frac{1}{\cos x} \]

These identities come from the definitions of sine, cosine, and tangent in a right-angled triangle or on the unit circle. They're especially useful because they express more complex ratios in terms of simpler sine and cosine functions. This simplification can significantly ease solving expressions, as seen in the original exercise where we used these identities to reframe the fraction before further simplification.

Even and Odd Functions

Understanding even and odd functions in trigonometry is crucial for simplifying expressions. These properties of trigonometric functions describe their symmetry and can help us manipulate them more conveniently.

• An even function satisfies the condition: \[ f(-x) = f(x) \] Examples include cos(x) and sec(x). This means their graphs are symmetric about the y-axis.
• An odd function satisfies: \[ f(-x) = -f(x) \] Sine and tangent are odd functions. Their graphs are symmetric about the origin.

In our exercise, recognizing that sec(x) is an even function enabled us to simplify the original expression: \[ \sec(-x) = \sec x \]This step considerably simplifies working with expressions, as it reduces the number of negative arguments we must handle.

Compound Fractions

Compound fractions can make trigonometric expressions look complicated, but they aren't so intimidating. Let's break down how to simplify them.A compound fraction (also known as a complex fraction) involves a fraction in either its numerator, its denominator, or both. Take for example the fraction:\[ \frac{\frac{\sin x}{\cos x}}{\frac{1}{\cos x}} \]Here, we have both the numerator and denominator as fractions.

• To simplify, multiply the numerator by the reciprocal of the denominator. In an instance like this, it becomes:\[ \frac{\sin x}{\cos x} \times \cos x \]
• This effectively cancels out the \( \cos x \) terms, leaving us with the much simpler expression:\[ \sin x \]

This is how we simplified our original trigonometric expression. Knowing this method can simplify many complicated-looking fractions due to their inclusion of another fraction in either the numerator or denominator.