Problem 79 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\frac{1}{\sec x+\tan x}+\frac{1}{\sec x-\tan x}=2 \sec x$$

Step-by-Step Solution

Answer

The given expression is equivalent to \(2 \sec x\), verifying the identity.

Step 1: Establish a Common Denominator

To simplify the given expression, let's start by establishing a common denominator for the two fractions. The denominators are \(\sec x + \tan x\) and \(\sec x - \tan x\). Multiply the numerators and denominators such that the common denominator is \[ (\sec x + \tan x)(\sec x - \tan x) = \sec^2 x - \tan^2 x \]We then have:\[ \frac{\sec x - \tan x}{\sec^2 x - \tan^2 x} + \frac{\sec x + \tan x}{\sec^2 x - \tan^2 x} \]

Step 2: Use Trigonometric Identities

Recall the trigonometric identity: \( \sec^2 x = 1 + \tan^2 x \). Then,\[ \sec^2 x - \tan^2 x = (1 + \tan^2 x) - \tan^2 x = 1 \]Therefore, the denominator simplifies to 1, and we are left with:\[ \sec x - \tan x + \sec x + \tan x \]

Step 3: Simplify the Numerator

Simplify the numerator by combining like terms:\[ \sec x - \tan x + \sec x + \tan x = 2 \sec x \]Thus, the given expression simplifies to:\[ \frac{1}{\sec x + \tan x} + \frac{1}{\sec x - \tan x} = 2 \sec x \]

Step 4: Verify the Identity

After simplifying both fractions and using the trigonometric identity, we have shown that the expression equals \(2 \sec x\). Thus, the original identity is verified:

Key Concepts

Verifying Trigonometric Identities

Verifying trigonometric identities involves showing that two different expressions are equivalent for all values that the variables can take. It's like solving a puzzle to prove that both sides of an equation, when simplified, are the same. This process requires familiarity with a variety of trigonometric identities, such as Pythagorean identities, reciprocal identities, and angle sum/difference identities.

Here's a simple approach to verification:

• Break down the more complex side or both sides of the equality, using known identities.
• Simplify them step by step.
• Gradually transform the expression until both sides match.

During this exercise, you are typically given one side more complex than the other. Your task is to manipulate it until it becomes identical to the simpler side, confirming that the identity holds true.

Common Denominators

When working with fractions, especially in trigonometric identities, finding a common denominator is crucial to simplify expressions. A common denominator allows us to combine fractions into a single expression.

Consider fractions of the form \(\frac{1}{a} + \frac{1}{b}\). To add these, we need a common denominator, \(a \times b\). We calculate as follows:

• Multiply the numerators and denominators to have a common base \(ab\).
• Combine them into a single fraction.
• Simplify if possible.

In the provided exercise, the denominators \( \sec x + \tan x \) and \( \sec x - \tan x \) were combined by multiplying to get \((\sec x + \tan x)(\sec x - \tan x) = \sec^2 x - \tan^2 x\). This is a standard approach whenever you have two fractions with different denominators and need to simplify them.

Trigonometric Simplification

Trigonometric simplification is the process of transforming complex trigonometric expressions into simpler forms using identities and algebraic manipulation. This can often involve using fundamental identities that link trigonometric functions.

For example, one important identity used frequently in simplification is \( \sec^2 x = 1 + \tan^2 x \). By substituting \( \sec^2 x - \tan^2 x \) with \(1\), we greatly simplified our expression from a complex fraction to simple terms.

Here are essential steps to simplify:

• Identify opportunities to use trigonometric identities to reduce complexity.
• Combine like terms and arrange terms to simplify it further.
• Ensure the final expression is in its simplest form, verifying all parts are reduced accurately through logical steps.

Through simplification, trigonometric expressions become far easier to manage, solving problems more efficiently and effectively.