Problem 18 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( \sin x \).

Step 1: Convert Secant and Tangent to Sine and Cosine

The expression given is \( \frac{\sec x - \cos x}{\tan x} \). Start by noting that \( \sec x = \frac{1}{\cos x} \) and \( \tan x = \frac{\sin x}{\cos x} \). Substitute these values into the expression. The expression now becomes \( \frac{\frac{1}{\cos x} - \cos x}{\frac{\sin x}{\cos x}} \).

Step 2: Simplify the Numerator

The numerator is \( \frac{1}{\cos x} - \cos x \). This can be rewritten as \( \frac{1 - \cos^2 x}{\cos x} \). Note that \( 1 - \cos^2 x = \sin^2 x \) using the Pythagorean identity. So, the numerator simplifies to \( \frac{\sin^2 x}{\cos x} \).

Step 3: Complete the Division

Now, substitute the simplified numerator back into the expression: \( \frac{\frac{\sin^2 x}{\cos x}}{\frac{\sin x}{\cos x}} \). Simplification by division results in \( \frac{\sin^2 x}{\cos x} \times \frac{\cos x}{\sin x} \).

Step 4: Simplify Further

The \(\cos x\) in the numerator and denominator cancel each other out, and \( \sin^2 x \) divided by \( \sin x \) simplifies to \( \sin x \). Therefore, the entire expression simplifies to \( \sin x \).

Key Concepts

Simplifying Expressions

When faced with the task of simplifying trigonometric expressions, the goal is to reduce the complexity of the expression using known identities and algebraic manipulations. This makes it easier to work with or evaluate. In trigonometry, simplification often involves substituting less familiar trigonometric functions like secant or tangent with their more familiar counterparts like sine and cosine.

• Start by identifying the trigonometric identities that fit your expression. This could involve substituting equivalent values.
• Combine similar terms whenever possible, and simplify the fractions by finding common denominators.
• Cancel out terms in the numerator and denominator to reduce to the simplest form.

In our example, transforming the secant and tangent into cosine and sine made the expression more straightforward, which is a key step in achieving simplification.

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric concept that is immensely useful in simplifying expressions. It states that: \[\sin^2 x + \cos^2 x = 1\]This identity helps us rewrite expressions in different forms and simplify them further. By rearranging the equation, we can derive other useful identities, such as:

• \(\sin^2 x = 1 - \cos^2 x\)
• \(\cos^2 x = 1 - \sin^2 x\)

In our exercise, the Pythagorean identity allowed us to replace \(1 - \cos^2 x\) with \(\sin^2 x\), which significantly simplified the numerator of the expression. It's an essential tool for converting and simplifying trigonometric expressions.

Trigonometric Functions

Trigonometric functions are the backbone of trigonometry, including sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), among others. These functions relate the angles of a triangle to the lengths of its sides and are pivotal in studying oscillations, waves, and circles. Each function has properties and identities associating them with one another, such as:

• \(\tan x = \frac{\sin x}{\cos x}\)
• \(\sec x = \frac{1}{\cos x}\)
• \(\csc x = \frac{1}{\sin x}\)

Understanding these relationships enables the simplification of expressions, as seen in our problem where secant and tangent were expressed in terms of sine and cosine. Learning how to convert between these functions aids in solving complex trigonometric equations and simplifying expressions systematically.