Problem 78 - 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. \( \dfrac{\tan x}{1 + \sec x} + \dfrac{1 + \sec x}{\tan x} \)

Step-by-Step Solution

Answer

The simplified form of the expression \( \dfrac{\tan x}{1 + \sec x} + \dfrac{1 + \sec x}{\tan x} \) is \( \dfrac{2 + \cos x}{\sin x (1 + \cos x)} \)

Step 1: Rewrite trigonometric expressions

Firstly, rewrite the expression using sine and cosine because they are the primary trigonometric functions and others can be expressed in terms of them. \nHere, \( \tan x = \dfrac{\sin x}{\cos x} \) and \( \sec x = \dfrac{1}{\cos x} \). So, substitute these values into the original expression: \( \dfrac{\dfrac{\sin x}{\cos x}}{1 + \dfrac{1}{\cos x}} + \dfrac{1 + \dfrac{1}{\cos x}}{\dfrac{\sin x}{\cos x}} \)

Step 2: Simplify the fractions

The next step is to simplify the complex fractions. To do that, multiply the numerator and the denominator of each fraction by \( \cos x \): \[ \dfrac{\sin x}{1 + \cos x} + \dfrac{\cos x + 1}{\sin x} \] Now, making the denominators the same for both fractions by multiplying the first fraction by \( \sin x \) and the second by \( 1 + \cos x \) can help to simplify the expression further.

Step 3: Perform addition operation

After step 2, the fractions' denominators are the same, allowing them to be easily added up. The result, after simplifying, will be: \[ \dfrac{\sin^2x + \cos x + 1}{\sin x (1 + \cos x)} \]

Step 4: Use Pythagorean identity to simplify

The Pythagorean identity \( \sin^2x + \cos^2x = 1 \) has been used to simplify the expression. The result is: \[ \dfrac{2 + \cos x}{\sin x (1 + \cos x)} \]

Key Concepts

Sine and Cosine

The sine and cosine functions are fundamental to understanding trigonometry. These functions relate to angles in right triangles. The sine of an angle, denoted as \( \sin x \), is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle. Conversely, the cosine of an angle, denoted as \( \cos x \), is the ratio of the adjacent side to the hypotenuse.

In situations where more complex trigonometric functions like tangent or secant appear, it's often useful to express them in terms of sine and cosine. For instance:

• \( \tan x = \dfrac{\sin x}{\cos x} \)
• \( \sec x = \dfrac{1}{\cos x} \)

Expressing these functions in terms of sine and cosine simplifies manipulation and understanding of trigonometric expressions.

Tangent and Secant

Tangent and secant are two important trigonometric functions that arise often in various problems. The tangent function, \( \tan x \), represents the ratio between the sine and cosine of an angle. Meanwhile, the secant function, \( \sec x \), is the reciprocal of the cosine function; it shows how far away you are from the cosine value equal to zero.

Here's why these functions are valuable:

• \( \tan x = \dfrac{\sin x}{\cos x} \), meaning when sine and cosine ratios are known, finding the tangent is straightforward.
• \( \sec x = \dfrac{1}{\cos x} \), allowing us to easily explore relationships involving the cosine function.

Using these relationships, one can transform complex expressions into simpler forms, aiding in the simplification process.

Pythagorean Identity

One of the most fundamental identities in trigonometry is the Pythagorean Identity. This identity involves the sine and cosine functions and states that for any angle \( x \), the following is always true: \( \sin^2 x + \cos^2 x = 1 \).

This identity is derived from the Pythagorean theorem in the context of the unit circle, where the radius of the circle is 1. The identity serves as a tool to simplify trigonometric expressions and verify other trigonometric identities. In our exercise, it allows us to replace \( \sin^2 x \) with \( 1 - \cos^2 x \), which can be incredibly useful when simplifying expressions.

Simplification of Trigonometric Expressions

Simplification of trigonometric expressions is a crucial skill in trigonometry. It involves using trigonometric identities to express complex expressions in a simpler or more convenient form. The process often includes steps such as common denominator finding, applying identities, and algebraic manipulation.

In the given exercise, we dealt with a complicated expression involving \( \dfrac{\tan x}{1 + \sec x} + \dfrac{1 + \sec x}{\tan x} \). By expressing tangent and secant in terms of sine and cosine, and using the Pythagorean identity, we could effectively simplify the given expression:

• Substituting \( \tan x \) and \( \sec x \) using their sine and cosine forms.
• Finding a common denominator for the complex fraction addition.
• Simplifying the result using the identity: \( \sin^2 x + \cos^2 x = 1 \).

The final simplified expression was \( \dfrac{2 + \cos x}{\sin x (1 + \cos x)} \). This demonstrates the power of strategic simplification in solving trigonometric expressions.