Problem 79 - Precalculus with Limits

Question

In Exercises 75 - 80, perform the addition or subtraction and use the fundamental identities to simplify. There is more than one correct form of each answer. \( \tan x + \dfrac{\cos x}{1 + \sin x} \)

Step-by-Step Solution

Answer

The simplified form of the given expression is \( \frac{1 + \sin x}{\cos x} \)

Step 1: Express tan(x) in terms of sine and cosine

Rewrite the given expression, replacing \( \tan x \) with \( \sin x / \cos x \) to obtain: \n \( \frac{\sin x}{\cos x} + \frac{\cos x}{1 + \sin x} \)

Step 2: Make denominators the same

To add the two fractions, we need to make their denominators the same. Therefore, rewrite the second fraction to have a denominator of \( \cos x (1 + \sin x) \). The expression will look like: \( \frac{\sin x}{\cos x} + \frac{\cos^2 x}{\cos x (1 + \sin x)} \)

Step 3: Simplify the expression

Using the Pythagorean identity \( \sin^2x + \cos^2x = 1 \), replace \( \cos^2x \) in the second fraction to get: \( \frac{\sin x}{\cos x} + \frac{1 - \sin^2 x}{\cos x (1 + \sin x)} \)

Step 4: Add the fractions and simplify further

Adding and simplifying the fractions, we get: \( \frac{\sin x (1 + \sin x) + 1 - \sin^2 x}{\cos x (1 + \sin x)} \) = \( \frac{1 + \sin x}{\cos x} \)

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in mathematics, particularly in the analysis of angles and the study of periodic phenomena. They relate the angles of a triangle to the lengths of its sides, commonly used in a variety of fields such as physics, engineering, and computer science.

• Sine, cosine, and tangent are the basic trigonometric functions.
• These functions are typically used with angles measured in degrees or radians.

Each of these functions serves a unique purpose in expressing different aspects of a right triangle.The tangent function, for example, is defined as the ratio of sine to cosine:\[\tan x = \frac{\sin x}{\cos x}.\] This relationship helps in expressing complex trigonometric identities and solves various problems involving triangles.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions that describe the shape of a right-angled triangle. For a given angle \( x \):

• Sine (\( \sin x \)) is defined as the ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos x \)) is defined as the ratio of the adjacent side to the hypotenuse.

In addition, these functions are crucial in relating angles and side lengths in trigonometric identities.The Pythagorean identity is one of the most important relationships involving sine and cosine:\[\sin^2 x + \cos^2 x = 1.\]This identity is extensively used for simplifying trigonometric expressions and solving equations.In the given exercise, replacing \( \tan x \) with its identity in terms of sine and cosine is a key step in simplifying the expression.

Addition and Subtraction of Fractions

The addition and subtraction of fractions involve finding a common denominator so that the numerators can be added or subtracted. This is a necessary step when dealing with trigonometric functions involving fraction forms.In our problem, we deal with two fractions:

• \( \frac{\sin x}{\cos x} \)
• \( \frac{\cos x}{1 + \sin x} \)

To add these fractions, we need to establish a common denominator, which is \( \cos x (1 + \sin x) \).Adjusting the fractions to have the same denominator allows for direct addition or subtraction of their numerators.The concept is simple:

• Multiply each term by whatever is necessary to achieve the same denominator.
• Perform the addition or subtraction on the numerators.
• Simplify the result.

This process will help simplify complex expressions in a step-by-step manner, making it easier to handle trigonometric identities.