Problem 15 - Precalculus Mathematics for Calculus

Question

Simplify the trigonometric expression. $$\frac{1+\cos y}{1+\sec y}$$

Step-by-Step Solution

Answer

The simplified expression is \(\cos y\).

Step 1: Substitute Secant with Cosine

Recall that the secant function is the reciprocal of the cosine function, i.e., \( \sec y = \frac{1}{\cos y} \). Substitute this into the expression to get:\[ \frac{1+\cos y}{1+\frac{1}{\cos y}} \]

Step 2: Simplify the Denominator

In the denominator, simplify the expression by combining into a single fraction:\[ 1 + \frac{1}{\cos y} = \frac{\cos y}{\cos y} + \frac{1}{\cos y} = \frac{\cos y + 1}{\cos y} \]

Step 3: Simplify the Overall Expression

Now the expression becomes:\[ \frac{1+\cos y}{\frac{\cos y + 1}{\cos y}} \]To divide by a fraction, multiply by its reciprocal:\[ \frac{1+\cos y}{1} \times \frac{\cos y}{\cos y + 1} \]The \(1+\cos y\) in the numerator can be canceled with the \(\cos y + 1\) in the denominator, as they are equal:\[ \frac{\cos y}{1} = \cos y \]

Step 4: Verify the Simplification

Ensure the expression is fully simplified. After canceling, we arrived at \( \cos y \), a well-known simplified trigonometric function.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is vital for simplifying complex expressions. These identities are equations that are true for all angles involved. Some common trigonometric identities include:

• Pythagorean identities, such as \( \sin^2 y + \cos^2 y = 1 \).
• Angle sum and difference identities, like \( \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \).
• Reciprocal identities, which relate the basic trigonometric functions to their reciprocals, such as \( \sec y = \frac{1}{\cos y} \).

When simplifying expressions, these identities help by converting one form into another, often making the expression more manageable. For example, by recognizing that \( \sec y \) can be rewritten as \( \frac{1}{\cos y} \), we can substitute directly in our given problem to make the expression easier to handle.

Reciprocal Function

In trigonometry, reciprocal functions play an integral role as they transform a primary trigonometric function into another. The key reciprocal functions are:

• Secant function: \( \sec y = \frac{1}{\cos y} \)
• Cosecant function: \( \csc y = \frac{1}{\sin y} \)
• Cotangent function: \( \cot y = \frac{1}{\tan y} \)

Using these transformations can often result in a simpler form. For example, considering the initial problem, substituting \( \sec y \) with \( \frac{1}{\cos y} \) allows for simplifying a complex fraction into a more straightforward one by combining terms or factoring where applicable. Recognizing when and how to use these reciprocal functions is fundamental in the process of expression simplification.

Expression Simplification

The process of simplification often involves several steps, allowing expressions to become more manageable. It usually involves:

• Identifying and applying known identities.
• Simplifying fractions and using common mathematical operations.
• Canceling out terms where possible to obtain a reduced form.

In the given exercise, after rewriting the expression using the reciprocal identity, we then simplify the fraction in the denominator to form a single fraction. Next, we multiply by its reciprocal to simplify our expression further. Canceling out identical expressions in the numerator and denominator reduces the fractions to their simplest form, as seen with \( \frac{1+\cos y}{1+\cos y} \), leaving \( \cos y \) behind.
The art of simplification requires patience and knowledge of the diverse identities and functions within trigonometry. Thus, it equips learners to tackle more complex mathematical problems with ease.