Problem 67 - A Graphical Approach to Precalculus with Limits

Question

Perform indicated operation and simplify the result. $$\frac{\cos x}{\sec x}+\frac{\sin x}{\csc x}$$

Step-by-Step Solution

Answer

The simplified result is 1.

Step 1: Rewrite Secant and Cosecant

First, recall that \( \sec x = \frac{1}{\cos x} \) and \( \csc x = \frac{1}{\sin x} \). Replace these in the expression: \[ \frac{\cos x}{\sec x} + \frac{\sin x}{\csc x} = \frac{\cos x}{\frac{1}{\cos x}} + \frac{\sin x}{\frac{1}{\sin x}} \].

Step 2: Simplify the Fractions

In each fraction, multiply the numerator by the reciprocal of the denominator: \[ \frac{\cos x}{\frac{1}{\cos x}} = \cos x \cdot \cos x = \cos^2x \] and \[ \frac{\sin x}{\frac{1}{\sin x}} = \sin x \cdot \sin x = \sin^2x \].

Step 3: Combine the Results

Add the two simplified terms together: \( \cos^2 x + \sin^2 x \).

Step 4: Apply the Pythagorean Identity

Recall the Pythagorean identity \( \cos^2 x + \sin^2 x = 1 \). Substitute this identity into the expression: \( 1 \).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry. It states that \( \cos^2 x + \sin^2 x = 1 \). This identity is derived from the Pythagorean theorem, which relates the sides of a right triangle. In a unit circle, where the radius is 1, this becomes especially useful.

The Pythagorean identity helps us simplify trigonometric expressions. It shows how the squares of the cosine and sine of any angle always add up to 1. Whenever you encounter \( \cos^2 x + \sin^2 x \) in an expression, you can replace it with 1, as seen in many trigonometric problems.

Remembering and applying this identity can make solving trigonometric expressions much more straightforward. If you find these terms in any mathematical context, think of the Pythagorean identity as your go-to tool for simplification.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves transforming a complex structure into a simpler one. You use identities, algebraic manipulation, and logical equations to achieve this.

Key steps include:

• Replacing complex functions with simpler identities.
• Combining like terms.
• Using algebraic operations to reduce the expression.

For instance, in the original problem, replacing \( \sec x \) with \( \frac{1}{\cos x} \) and \( \csc x \) with \( \frac{1}{\sin x} \) allows you to write the expression in terms of basic sine and cosine functions. Simplifying fractions by multiplying by the reciprocal can quickly lead to a much more manageable form.

Moreover, recognizing patterns such as the Pythagorean identity can swiftly collapse multiple terms into a single, straightforward result. This not only simplifies your work but ensures accuracy in your final answer.

Trigonometric Functions

Trigonometric functions are functions of angles and are fundamental in the field of trigonometry. The basic functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). There are also what we call reciprocal trigonometric functions: secant (\( \sec \)), cosecant (\( \csc \)), and cotangent (\( \cot \)).

Each of these functions corresponds to ratios of sides in a right triangle:

• \( \sin x = \text{opposite/hypotenuse} \)
• \( \cos x = \text{adjacent/hypotenuse} \)
• \( \tan x = \text{opposite/adjacent} \)
• \( \sec x = \frac{1}{\cos x} \)
• \( \csc x = \frac{1}{\sin x} \)
• \( \cot x = \frac{1}{\tan x} \)

Understanding these ratios helps with solving numerous mathematical problems involving angles. They are widely used not just in geometry, but also in fields like physics, engineering, and computer science. Simplifying expressions often involves converting between these functions to find the most efficient solution.