Problem 71 - A Graphical Approach to Precalculus with Limits

Question

Perform indicated operation and simplify the result. $$\frac{1}{1+\cos x}-\frac{1}{1-\cos x}$$

Step-by-Step Solution

Answer

The simplified result is \( \frac{-2\cos x}{\sin^2 x} \).

Step 1: Identify the Problem

The given expression requires simplification.\[\frac{1}{1+\cos x} - \frac{1}{1-\cos x}\] The task involves performing the subtraction of two fractions.

Step 2: Find a Common Denominator

To subtract these fractions, we need a common denominator. The common denominator is the product of the two individual denominators:\[(1+\cos x)(1-\cos x)\]

Step 3: Simplify the Common Denominator

Use the identity for the difference of squares to write the common denominator as:\[1 - \cos^2 x = \sin^2 x\] Hence, the new denominator becomes \( \sin^2 x \).

Step 4: Rewrite Each Fraction with the Common Denominator

Rewrite each fraction over the common denominator \( \sin^2 x \):\[\frac{1}{1+\cos x} = \frac{1-\cos x}{\sin^2 x}, \quad \frac{1}{1-\cos x} = \frac{1+\cos x}{\sin^2 x}\]

Step 5: Perform the Subtraction

Subtract the fractions using the common denominator:\[\frac{1-\cos x}{\sin^2 x} - \frac{1+\cos x}{\sin^2 x} = \frac{(1-\cos x) - (1+\cos x)}{\sin^2 x}\]Simplify the numerator:\[(1-\cos x) - (1+\cos x) = 1 - \cos x - 1 - \cos x = -2\cos x\]

Step 6: Simplify the Result

Substitute the simplified numerator back over the common denominator:\[\frac{-2\cos x}{\sin^2 x}\]This is the simplified form of the original expression.

Key Concepts

Fraction Simplification

When simplifying fractions like \[\frac{1}{1+\cos x} - \frac{1}{1-\cos x},\]our goal is to make the expression more manageable by converting it into a simplified form. For this particular exercise, the simplification process involves several steps to achieve a cleaner result.

• First, you identify the entire expression as a fraction with two parts (or terms) that will be subtracted.
• Next, you simplify each part by finding a common denominator, making subtraction possible.
• Finally, combine the parts and simplify as much as possible, leaving it in a form that's easy to understand and further calculate if needed.

Simplifying fractions is a crucial skill in algebra and trigonometry because it provides a clearer view of the mathematical relationships and allows for more straightforward manipulation in future steps.

Common Denominator

To subtract two fractions as seen in\[\frac{1}{1+\cos x} - \frac{1}{1-\cos x},\]we need a common denominator. This step is vital because it allows us to combine the fractions into a single expression.
Here's how the common denominator is found:

• Each original fraction has a denominator: \(1+\cos x\) for the first and \(1-\cos x\) for the second.
• Multiply these denominators together to establish a common denominator, resulting in \((1+\cos x)(1-\cos x)\).

Using the difference of squares identity, which is \[a^2 - b^2 = (a-b)(a+b),\]we can simplify \((1+\cos x)(1-\cos x)\) to \(1-\cos^2 x\),which is equal to \(\sin^2 x\).
This simplification means both fractions can be expressed over the common denominator \(\sin^2 x\).

Trigonometric Functions

Trigonometric functions such as sine and cosine play a key role in this problem. The given fraction uses the cosine function, \(\cos x\), within its denominators initially.
Key trigonometric identities help simplify expressions massively:

• The identity \(1-\cos^2 x = \sin^2 x\) is crucial here. Known as the Pythagorean identity, it allows conversion of the expression to involve \(\sin^2 x\) instead.
• Simplified expressions like \(\frac{-2\cos x}{\sin^2 x}\) involve recognizing how these trigonometric functions interact with each other through identities.

Understanding trigonometric functions means becoming familiar with their behaviors and relationships, you'll often encounter these in manipulating and simplifying expressions in practice.