Problem 19 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Common Denominator

To simplify the expression, we first look for a common denominator for the two fractions. The common denominator for \( \cos u \) and \( 1 + \sin u \) is \( \cos u(1 + \sin u) \).

Step 2: Rewrite Each Fraction

Re-write each fraction using the common denominator. The first fraction \( \frac{1 + \sin u}{\cos u} \) becomes: \[ \frac{(1 + \sin u)^2}{\cos u(1 + \sin u)} \]. The second fraction \( \frac{\cos u}{1 + \sin u} \) becomes: \[ \frac{\cos^2 u}{(1 + \sin u)\cos u} \].

Step 3: Combine the Fractions

Now, combine these fractions over their common denominator: \[ \frac{(1 + \sin u)^2 + \cos^2 u}{\cos u(1 + \sin u)} \].

Step 4: Expand and Simplify the Numerator

Expand \((1 + \sin u)^2\): \((1 + \sin u)^2 = 1 + 2\sin u + \sin^2 u\). So the numerator becomes: \[ 1 + 2\sin u + \sin^2 u + \cos^2 u \].

Step 5: Apply Pythagorean Identity

Use the identity \( \sin^2 u + \cos^2 u = 1 \) to simplify the numerator: \[ 1 + 2\sin u + (\sin^2 u + \cos^2 u) = 1 + 2\sin u + 1 = 2 + 2\sin u \].

Step 6: Factor and Simplify

Factor the numerator: \( 2 + 2\sin u = 2(1 + \sin u) \). The expression becomes: \[ \frac{2(1 + \sin u)}{\cos u (1 + \sin u)} \].

Step 7: Cancel Common Terms

Cancel \( 1 + \sin u \) from the numerator and the denominator: \[ \frac{2}{\cos u} \].

Step 8: Final Expression

The expression simplifies to \( \frac{2}{\cos u} \). This is equivalent to \( 2\sec u \).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships between trigonometric functions that are always true for any angle. They are essential tools in trigonometry, helping us simplify and manipulate trigonometric expressions efficiently. These identities include relationships like the reciprocal identities, the quotient identities, and the Pythagorean identities. Each identity offers a different perspective on how the trigonometric functions relate to each other.

When tackling trigonometric expressions, these identities can significantly reduce complexity by transforming the expressions into a more manageable form. For example, converting \(\sec u\) into \(\frac{1}{\cos u}\) or simplifying the sum \(\sin^2 u + \cos^2 u\) using the Pythagorean identity. Familiarity with these trigonometric identities allows you to recognize patterns and opportunities to simplify your work with trigonometric expressions.

• Reciprocal Identities (e.g., \(\sec u = \frac{1}{\cos u}\))
• Pythagorean Identities (e.g., \(\sin^2 u + \cos^2 u = 1\))
• Quotient Identities (e.g., \(\tan u = \frac{\sin u}{\cos u}\))

Pythagorean Identity

One of the most commonly used trigonometric identities is the Pythagorean identity, \(\sin^2 u + \cos^2 u = 1\). This identity derives from the Pythagorean theorem and is central in simplifying trigonometric expressions. It shows the intrinsic link between the sine and cosine functions.

In the original exercise solution, the Pythagorean identity makes an appearance to help simplify the complex expression. By recognizing that \(\sin^2 u + \cos^2 u\) equals 1, it enables a major simplification in the expression from one that might be difficult to handle to something much simpler. Recognizing this identity allows you to see beyond the superficial complexity of expressions and transform them easily.

• Helps in simplifying expressions like \(1 + \sin^2 u + \cos^2 u\)
• Enables transitions from complex expressions to simpler, factorable forms
• Derived from the Pythagorean theorem in geometry

Simplifying Fractions

Simplifying fractions in trigonometric expressions helps make calculations easier and the expressions more understandable. When dealing with complex expressions, identifying a common denominator is often the first step. This allows you to merge multiple fractions into a single fraction.

In the exercise solution, simplifying fractions involved identifying the common denominator \(\cos u(1 + \sin u)\) for the given fractions. By rewriting each fraction with this common denominator, you can combine them into one. Once combined, it becomes much simpler to tackle further simplifications like expansion and applying known identities.

• Combine multiple fractions into one with a common denominator
• Make use of identities to simplify the expression further
• Cancel out common terms to achieve the simplest form

Simplifying fractions not only aids in solving exercises but also improves your understanding of how trigonometric expressions interact and transform.