Problem 21 - Precalculus Mathematics for Calculus

Question

Simplify the trigonometric expression. $$\frac{1+\sin u}{\cos u}+\frac{\cos u}{1+\sin u}$$

Step-by-Step Solution

Answer

The expression simplifies to \(2 \sec u\).

Step 1: Identify a Common Denominator

To simplify the expression, first find a common denominator for the fractions \(\frac{1+\sin u}{\cos u}\) and \(\frac{\cos u}{1+\sin u}\). The least common denominator (LCD) is \(\cos u (1+\sin u)\).

Step 2: Rewrite Each Fraction Over the Common Denominator

Rewrite each fraction to have the common denominator \(\cos u (1+\sin u)\):\[\frac{1+\sin u}{\cos u} = \frac{(1+\sin u)^2}{\cos u (1+\sin u)}\]\[\frac{\cos u}{1+\sin u} = \frac{\cos^2 u}{\cos u (1+\sin u)}\]

Step 3: Combine the Fractions

Combine the two fractions into a single fraction:\[\frac{(1+\sin u)^2 + \cos^2 u}{\cos u (1+\sin u)}\]

Step 4: Simplify the Numerator

Expand and simplify the numerator. First, expand \((1+\sin u)^2 = 1 + 2\sin u + \sin^2 u\). Now add \(\cos^2 u\):\[1 + 2\sin u + \sin^2 u + \cos^2 u\]Since \(\sin^2 u + \cos^2 u = 1\), substitute 1 for \(\sin^2 u + \cos^2 u\):\[1 + 2\sin u + 1 = 2 + 2\sin u\]

Step 5: Simplify the Final Expression

The simplified numerator is \(2(1+\sin u)\). Therefore, the expression becomes:\[\frac{2(1+\sin u)}{\cos u(1+\sin u)}\]Cancel \(1+\sin u\) in the numerator and denominator (assuming \(1+\sin u eq 0\)) to get:\[\frac{2}{\cos u} = 2 \sec u\]

Step 6: Therefore, the Simplified Expression is

The simplified trigonometric expression is \(2 \sec u\).

Key Concepts

Simplifying Fractions

Fractions can be complex, especially when dealing with trigonometric expressions. Simplifying them can help make calculations easier and the results more meaningful. Let's take the example given and break it down into simple steps.

Imagine you have an expression with two fractions, and your goal is to reduce them to a simpler form. When simplifying, one strategy is to find ways to combine terms, numerator and denominator, and reduce redundant elements.

• Identify parts of the fraction that are common or look for trigonometric identities that can simplify parts of the fraction.
• Divide or cancel similar terms in the numerator and the denominator to shrink the fraction.
• If a common factor, like \(1+\sin u\), appears in both the numerator and the denominator, it can be cancelled out under certain conditions.

Simplifying fractions in trigonometry often involves using identities or algebraic manipulation. Cancellation is a handy tool but must be used correctly to avoid inaccuracies!

Common Denominator

When adding or subtracting fractions, having a common denominator is crucial. The common denominator allows you to combine the fractions more easily.

For the original problem, \(\frac{1+\sin u}{\cos u} + \frac{\cos u}{1+\sin u}\), the goal was to find a way to express both fractions with the same base. The least common denominator (LCD) becomes evident as \(\cos u (1+\sin u)\).

• Find a value that both denominators can evenly multiply into; this is your common denominator.
• Rewrite each fraction so they share this new denominator, making the addition or subtraction of the fractions a straightforward task.
• Ensure the numerators are correctly adjusted to maintain the equivalence of the original fractions.

With the common denominator determined, it’s much easier to add the fractions by combining their adjusted numerators.

Trigonometric Identities

Trigonometric identities are essential tools for simplifying expressions. They allow you to replace parts of an equation with equal values that might be easier to work with.

In the given problem, the identity \(\sin^2 u + \cos^2 u = 1\) was used to simplify the numerator of the combined fraction. Factoring in trigonometric identities can simplify complex expressions considerably.

• Recognize patterns in the given problem that match trigonometric identities; doing so allows you to switch expressions for their identities.
• Apply the identities step by step, making the trigonometric expression smoother and easier to handle.
• The use of identities is critical in transforming \(1 + 2\sin u + \sin^2 u + \cos^2 u\) into \(2 + 2\sin u\), leveraging that \(\sin^2 u + \cos^2 u = 1\).

Mastering these identities unlocks a great deal of manipulation potential in trigonometric problems, helping transform complex expressions into their simplest form.