Problem 22 - Algebra and Trigonometry

Question

Simplify the trigonometric expression. $$ \tan x \cos x \csc x $$

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Understand the Problem

We are asked to simplify the given trigonometric expression: \( \tan x \cos x \csc x \). This involves reducing the expression to its simplest form using trigonometric identities.

Step 2: Express Trigonometric Functions in Basic Terms

We start by expressing each trigonometric function in terms of sine and cosine:1. \( \tan x = \frac{\sin x}{\cos x} \)2. \( \csc x = \frac{1}{\sin x} \)3. \( \cos x = \cos x \) remains as it is.

Step 3: Substitute the Expressions

Substitute the expressions for \( \tan x \) and \( \csc x \) in the original expression:\( \tan x \cos x \csc x = \left( \frac{\sin x}{\cos x} \right) \cos x \left( \frac{1}{\sin x} \right) \).

Step 4: Simplify the Expression

Simplify the expression by canceling out the common terms from numerator and denominator:1. \( \cos x \) in the numerator and \( \cos x \) in the denominator cancel out.2. \( \sin x \) in the numerator and \( \sin x \) in the denominator cancel out.This leaves us with the expression simplified to \( 1 \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that relate different trigonometric functions to one another. These identities are helpful in simplifying trigonometric expressions, proving equations, and solving trigonometric equations. The key to using trigonometric identities is to recognize patterns and express complex functions in terms of these simpler basic identities.

• Basic Identities: These include \( \sin^2 x + \cos^2 x = 1 \), which is known as the Pythagorean identity. It demonstrates the intrinsic relationship between sine and cosine functions.
• Quotient Identities: The identity \( \tan x = \frac{\sin x}{\cos x} \) expresses tangent in terms of sine and cosine. This identity is essential for many simplification processes because it helps transform expressions involving tangent into ones that can be more easily combined or canceled.
• Reciprocal Identities: These include \( \csc x = \frac{1}{\sin x} \) and \( \sec x = \frac{1}{\cos x} \). They express trigonometric functions as reciprocals, which is helpful for simplifications and calculations.

In the given exercise, by recognizing and applying quotient and reciprocal identities, the expression was simplified effectively.

Sine and Cosine Functions

Sine and cosine are the foundational trigonometric functions in mathematics, representing the primary ratios in a right-angled triangle. They are defined using the unit circle or triangle properties and are periodic functions with a period of \(2\pi\). Their importance in trigonometric expressions lies in their ability to express other trigonometric functions in terms of these two functions.

• Sine Function (\( \sin x \)): Represents the ratio of the opposite side to the hypotenuse in a right-angled triangle. It is defined as \( \sin x = \frac{\text{Opposite}}{\text{Hypotenuse}} \).
• Cosine Function (\( \cos x \)): Represents the ratio of the adjacent side to the hypotenuse, and is defined as \( \cos x = \frac{\text{Adjacent}}{\text{Hypotenuse}} \).

In the problem, transforming expressions such as \( \tan x \) and \( \csc x \) into sine and cosine forms makes the process of simplification more straightforward, as it facilitates the cancellation of terms and simplification of complex equations.

Simplifying Expressions

Simplifying trigonometric expressions is the process of reducing them to their simplest form. This is achieved by replacing complex trigonometric terms with simpler or standard terms using identities and arithmetical operations. A well-simplified expression is easier to compute or further manipulate in solving equations.To simplify the expression \( \tan x \cos x \csc x \):

• First, recognize and use appropriate trigonometric identities to rewrite terms: \( \tan x = \frac{\sin x}{\cos x} \) and \( \csc x = \frac{1}{\sin x} \).
• Substitute these identities into the expression to transform it into \( \left( \frac{\sin x}{\cos x} \right) \cos x \left( \frac{1}{\sin x} \right) \).
• Simplify by canceling out common terms: \( \cos x \) and \( \sin x \) appear in both the numerator and denominator, allowing them to be canceled.

This results in the entire expression simplifying to \( 1 \). Simplification not only reduces calculation complexity but also prepares the expression for further mathematical operation if needed.