Problem 48 - Algebra 2

Question

Simplify each expression. $$ \frac{\cos 2 \theta}{\sin \theta+\cos \theta} $$

Step-by-Step Solution

Answer

The given expression cannot be simplified further, its simplest form is \(\frac{\cos 2\theta}{\sin \theta+\cos \theta}\)

Step 1: Apply double-angle identity

Use the double angle identity \(\cos 2\theta = \cos^2\theta - \sin^2\theta\), the expression becomes \(\frac{\cos^2\theta - \sin^2\theta}{\sin\theta + \cos\theta}\)

Step 2: Rewrite with Pythagorean identity

Since we know that \(\sin^2\theta + \cos^2\theta = 1\), we can write \(\sin^2\theta\) as \(1 - \cos^2\theta\). The expression then changes to \(\frac{\cos^2\theta - (1 - \cos^2\theta)}{\sin\theta + \cos\theta}\) which simplifies to \(\frac{2\cos^2\theta - 1}{\sin\theta + \cos\theta}\)

Step 3: Applying double-angle identity again

This time round, we can transform the numerator's expression into \(\cos 2\theta\). The expression becomes \(\frac{\cos 2\theta}{\sin\theta + \cos\theta}\)

Step 4: Remove common terms

Seeing that the expression rewrites to what we had at the start, conclude that the expression can't be simplified further because we're stuck in a loop back to the original expression

Key Concepts

Double Angle Identities

In trigonometry, double angle identities are extremely useful tools for transforming and simplifying expressions. A double angle identity essentially relates the values of trigonometric functions at double an angle to those at the original angle. One of the commonly used double angle identities is for cosine, given by:

• \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)

This identity is particularly advantageous as it allows us to express the cosine of twice an angle in terms of the squares of sine and cosine of the angle itself. This transformation is often the first step in simplifying tricky trigonometric expressions.
In our problem, we employed this identity to alter the initial expression \( \frac{\cos 2\theta}{\sin \theta + \cos \theta} \) into \( \frac{\cos^2 \theta - \sin^2 \theta}{\sin \theta + \cos \theta} \).
Grasping how to apply this identity is crucial for resolving many trigonometric challenges as it often unveils simpler forms of the same expression.

Pythagorean Identity

The Pythagorean identity is one of the foundational trigonometric identities. It connects the squares of sine and cosine functions to the number one. The identity is expressed as:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This simple yet powerful identity acts as a building block for many trigonometric simplifications and proofs.
In the context of our problem, after applying the double angle identity, we turned to the Pythagorean identity to rewrite \( \sin^2 \theta \) as \( 1 - \cos^2 \theta \). Doing this enabled us to proceed with simplifying the expression further:

• \( \cos^2 \theta - (1 - \cos^2 \theta) = 2\cos^2 \theta - 1 \)

This kind of manipulation highlights the importance of being comfortable with re-expressing terms in equations using fundamental identities like the Pythagorean identity.
It is a critical step that often makes the difference between complex, cumbersome expressions and their simpler versions.

Expression Simplification

Expression simplification in trigonometry is all about recognizing patterns and knowing which identities and algebraic techniques to apply at each step. Initially, the goal is to massage a complex-looking expression into a form that is easier to understand or resolve.
In our illustrative example, even though we initially simplified the expression, the process showed us that some expressions may simply loop back to their starting point—essentially showing that they can't be simplified.
A key part of simplification is getting accustomed to testing different forms of identities:

• Use double angle and Pythagorean identities generously.
• Re-organize or decompose parts of the expression.
• Identify repeated transformations that return to the initial forms.

Though the process might lead back to the original form, as it did in our example, it serves as an important exercise in analysis and problem-solving techniques.
In summary, mastering simplification involves patience and practice with numerous expressions. Understanding that not all expressions have a straightforward simplified version, but the exercise broadens your problem-solving skills significantly.