Problem 14 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recall Trigonometric Identities

First, we need to recall that \( \sec(-x) = \frac{1}{\cos(-x)} \) and \( \tan x = \frac{\sin x}{\cos x} \). The even property of cosine shows \( \cos(-x) = \cos x \), hence \( \sec(-x) = \frac{1}{\cos x} = \sec x \).

Step 2: Rewrite the Expression

Substitute the identities into the original expression: \( \frac{\tan x}{\sec (-x)} \). Using \( \sec(-x) = \frac{1}{\cos x} \), we rewrite the expression as \( \frac{\tan x}{\sec x} \) which simplifies to \( \tan x \cdot \cos x \) since \( \sec x = \frac{1}{\cos x} \).

Step 3: Simplify Further

Substitute \( \tan x = \frac{\sin x}{\cos x} \) into the expression: \( \frac{\sin x}{\cos x} \cdot \cos x \). The \( \cos x \) in the numerator and the denominator cancels out, leaving \( \sin x \).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in simplifying complex expressions and solving trigonometric equations. They are equations that involve trigonometric functions and are true for every value of the occurring variables. Some common trigonometric identities include:

• Reciprocal identities: like \( \sin x = \frac{1}{\csc x} \) and \( \sec x = \frac{1}{\cos x} \).
• Quotient identities: such as \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \).
• Pythagorean identities: including \( \sin^2 x + \cos^2 x = 1 \).

These identities help to transform a given expression into a more manageable form. For the example in the original exercise, knowing that \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \) allows simplification by expressing different functions in terms of sine and cosine.
Once expressions have a common form, terms can often be cancelled or factored out, significantly simplifying the original task.

Even-Odd Identities

Understanding even-odd identities is crucial in the context of trigonometric simplification. An even function is symmetric about the y-axis, while an odd function is symmetric about the origin. In trigonometry:

• The cosine function \( \cos(-x) = \cos x \) is even, which means it doesn't change its value when replacing \( x \) with \( -x \).
• The sine \( \sin(-x) = -\sin x \) and tangent \( \tan(-x) = -\tan x \) functions are odd, reflecting their symmetry properties.

In the provided simplification problem, we use the fact that \( \sec(-x) = \frac{1}{\cos(-x)} \), and since cosine is even, \( \cos(-x) = \cos x \). This becomes \( \sec x \), helping to streamline the expression. Understanding these identities helps to predict how changes in signs affect trigonometric expressions, thus enabling further simplification.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves converting them into simpler or more familiar forms using known identities and algebraic manipulation. The goal is to transform the expression into its simplest possible form, often reducing it to a single trigonometric function or a straightforward numeral. In the exercise, the original expression \( \frac{\tan x}{\sec(-x)} \) is simplified effectively by:

• Applying trigonometric identities like \( \sec(-x) = \sec x \) using the properties of even functions.
• Replacing \( \tan x \) with \( \frac{\sin x}{\cos x} \) and cancelling common terms.

This practice simplifies the expression to \( \sin x \), making calculations much easier. Simplifying expressions is a skill that grows with practice. It boosts your understanding of how trigonometric functions relate, allowing for efficient problem-solving. Keeping these strategies in mind can aid tremendously in both academic and real-world applications.