Problem 142 - Calculus Volume 1

Question

Simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. \(\sec x-\cos x\)

Step-by-Step Solution

Answer

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

Step 1: Convert Secant to Cosine

Recall that secant is the reciprocal of cosine. Therefore, \( \sec x = \frac{1}{\cos x} \). Substitute \( \sec x \) in the expression to get: \( \frac{1}{\cos x} - \cos x \).

Step 2: Simplify the Expression

To simplify \( \frac{1}{\cos x} - \cos x \), we need a common denominator. The common denominator for \( \frac{1}{\cos x} \) and \( \cos x \) is \( \cos x \). Rewrite the expression as \( \frac{1}{\cos x} - \frac{\cos^2 x}{\cos x} \).

Step 3: Combine the Fractions

With a common denominator, combine the fractions: \( \frac{1 - \cos^2 x}{\cos x} \).

Step 4: Use Trigonometric Identity

Use the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \). Thus, \( 1 - \cos^2 x = \sin^2 x \). Substitute this into the expression: \( \frac{\sin^2 x}{\cos x} \).

Step 5: Simplify Further

Recognize that \( \frac{\sin^2 x}{\cos x} = \frac{\sin x \cdot \sin x}{\cos x} \). This can also be written as \( \sin x \cdot \tan x \), since \( \tan x = \frac{\sin x}{\cos x} \). So the simplified expression is \( \sin x \cdot \tan x \).

Key Concepts

Secant Function

The secant function, represented as \( \sec x \), is one of the six primary trigonometric functions commonly used in mathematics. It is the reciprocal function of the cosine. This means wherever we have the secant function, it can be expressed as \( \sec x = \frac{1}{\cos x} \). This relationship is fundamental in simplifying trigonometric expressions.

Understanding this reciprocal relationship:

• Helps transform complicated expressions, making them easier to work with.
• Allows expressions involving secant to be written in terms of cosine, which is often more manageable for calculations.

By converting \( \sec x \) into \( \frac{1}{\cos x} \), we then proceed to solve trigonometric problems by utilizing the simpler terms of sine and cosine functions.

Pythagorean Identity

One of the most essential tools in trigonometry is the Pythagorean Identity. This identity states that \( \sin^2 x + \cos^2 x = 1 \). This theorem holds true for all angles \( x \) and is derived from the Pythagorean theorem applied to a right triangle.

This identity

• Allows conversion between sine and cosine values easily.
• Is extremely useful in rewriting and simplifying trigonometric expressions, as it provides a way to swap between \( \sin^2 x \) and \( \cos^2 x \).

When simplifying an expression like \( 1 - \cos^2 x \), it’s convenient to recognize that it equals \( \sin^2 x \) using this identity. This transformation is crucial for further simplification in trigonometric solutions.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves using identities and algebraic manipulations to present the expression in a simpler or more standard form. This often means expressing functions in terms of basic trigonometric functions like sine and cosine before proceeding.

Steps for simplification include:

• Converting functions such as tangent, secant, and others into terms of sine and cosine.
• Finding common denominators to combine fractions efficiently.
• Utilizing identities like the Pythagorean Identity to transform the expression.

For instance, the simplification of \( \frac{1 - \cos^2 x}{\cos x} \) uses the identity \( 1 - \cos^2 x = \sin^2 x \). The expression is then broken down to a form such as \( \sin x \cdot \tan x \), where \( \tan x \) is \( \frac{\sin x}{\cos x} \). Each step ensures the expression is made as compact and understandable as possible.