Problem 19 - Precalculus Mathematics for Calculus

Question

Simplify the trigonometric expression. $$\frac{1+\csc x}{\cos x+\cot x}$$

Step-by-Step Solution

Answer

The expression simplifies to \(\sec x\).

Step 1: Express in terms of basic trigonometric functions

We begin by rewriting the expression using basic trigonometric functions. Recall that \(\csc x = \frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\). Substitute these into the original expression:\[\frac{1+\csc x}{\cos x + \cot x} = \frac{1 + \frac{1}{\sin x}}{\cos x + \frac{\cos x}{\sin x}}.\]

Step 2: Combine fractions in the numerator

Combine the terms in the numerator. To combine \(1\) and \(\frac{1}{\sin x}\), we use a common denominator:\[1 + \frac{1}{\sin x} = \frac{\sin x}{\sin x} + \frac{1}{\sin x} = \frac{\sin x + 1}{\sin x}.\]

Step 3: Combine fractions in the denominator

Now, combine the terms in the denominator. We have \(\cos x + \frac{\cos x}{\sin x}\). Using a common denominator gives us:\[\cos x + \frac{\cos x}{\sin x} = \frac{\cos x \cdot \sin x}{\sin x} + \frac{\cos x}{\sin x} = \frac{\cos x \cdot \sin x + \cos x}{\sin x} = \frac{\cos x(\sin x + 1)}{\sin x}.\]

Step 4: Simplify the expression

Now that we have common denominators, we can rewrite the expression:\[\frac{\sin x + 1}{\sin x} \div \frac{\cos x(\sin x + 1)}{\sin x}.\]When we divide fractions, we multiply by the reciprocal:\[\frac{\sin x + 1}{\sin x} \times \frac{\sin x}{\cos x(\sin x + 1)} = \frac{(\sin x + 1) \cdot \sin x}{\cos x (\sin x + 1) \cdot \sin x}.\]The \(\sin x\) in the numerator and denominator cancel, and so do \(\sin x + 1\), leaving:\[\frac{1}{\cos x} = \sec x.\]

Step 5: Conclusion

The given trigonometric expression simplifies to \(\sec x\). Thus, the original expression reduces to the secant function of \(\cos x\).

Key Concepts

Basic Trigonometric Identities

Trigonometric identities are like a toolkit for working with angles and sides in a triangle, particularly in the context of a right triangle. These identities help us understand the relationships between the different trigonometric functions. Some of the most common identities are:

• Sine and cosine are fundamental, defining relationships between the angles and sides of a triangle.
• The reciprocal identities present another layer: for example, \( \csc x = \frac{1}{\sin x} \) and \( \sec x = \frac{1}{\cos x} \).
• Other helpful identities include tangent and cotangent, where \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \).

These identities are used to simplify complex expressions, like the one in the exercise. By substituting these identities into an expression, calculations become easier, and we can reveal hidden equalities or further simplify our expressions. Developing a good grasp of these identities enables students to work more efficiently with trigonometric problems.

Trigonometric Functions

Trigonometric functions are essential in understanding angles and their relationships in various fields, from physics to geometry. These functions—sine, cosine, tangent, cosecant, secant, and cotangent—are ratios of different sides of a right triangle.

• Sine (\( \sin \)): the ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos \)): the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \)): the ratio of the opposite side to the adjacent side.
• Cosecant (\( \csc \)): the reciprocal of sine.
• Secant (\( \sec \)): the reciprocal of cosine.
• Cotangent (\( \cot \)): the reciprocal of tangent.

Understanding these functions is crucial as they form the basis for simplifying and transforming expressions in trigonometry. They help in solving equations and proving identities, providing us a pathway to analyze different geometric problems effectively.

Secant Function

The secant function, denoted as \( \sec x \), is one of the reciprocal trigonometric functions. It's defined as the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \).
The significance of the secant function emerges in various practical applications. It's particularly useful when dealing with angles greater than 90 degrees, where cosine and other functions might be less intuitive to work with.

• In many instances, such as simplifying expressions, introducing \( \sec x \) provides a simpler or more coherent result.
• Knowing how to switch between \( \cos x \) and \( \sec x \) enhances our problem-solving abilities, enabling more streamlined solutions.

In our exercise, transforming the expression to \( \sec x \) illustrates its utility in revealing a simpler form. This process underscores the importance of understanding and applying the secant function in solving trigonometric problems effectively.