Problem 22 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The simplified trigonometric expression is 1.

Step 1: Understand the Expression

We start with the given trigonometric expression \( \tan x \cos x \csc x \). We need to simplify this expression using trigonometric identities.

Step 2: Identify Trigonometric Identities

Recall the trigonometric identities:1. \( \tan x = \frac{\sin x}{\cos x} \)2. \( \csc x = \frac{1}{\sin x} \)We will use these identities to rewrite the expression in terms of sine and cosine.

Step 3: Apply Trigonometric Identities

Substitute the identities into the 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

Cancel the \( \cos x \) in the numerator with the \( \cos x \) in the denominator, and cancel the \( \sin x \) in the numerator with \( \sin x \) in the denominator. \[ \frac{\sin x}{\cos x} \cdot \cos x \cdot \frac{1}{\sin x} = 1 \]

Step 5: Conclusion

After simplifying, we have found that the entire expression simplifies to 1.

Key Concepts

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves breaking down complex trigonometric formulas into simpler, more manageable parts. This often includes using trigonometric identities to rewrite and simplify expressions.
These identities help to change the form of the expression without changing its value.

To simplify the expression \( \tan x \cos x \csc x \), recognize the following basic trigonometric identities:

• \( \tan x = \frac{\sin x}{\cos x} \)
• \( \csc x = \frac{1}{\sin x} \)

Replace \( \tan x \) and \( \csc x \) in the original expression using these identities. This transforms the expression into:

\[ \frac{\sin x}{\cos x} \cdot \cos x \cdot \frac{1}{\sin x} \]

By carefully canceling out the common terms of \( \cos x \) and \( \sin x \), we simplify the expression to 1. This process showcases the power and utility of trigonometric identities in simplifying complex expressions.

Basic Trigonometry

Basic trigonometry lays the foundation for understanding angles and the relationships between the sides of triangles. It is fundamental in the field of mathematics and widely used in various applications.
It involves basic trigonometric functions, identities, and formulas.

Here are the key trigonometric functions commonly used:

• Sine \( \sin x \)
• Cosine \( \cos x \)
• Tangent \( \tan x \)
• Cosecant \( \csc x \)
• Secant \( \sec x \)
• Cotangent \( \cot x \)

These functions are defined for any angle \( x \) and relate the angles of a triangle to ratios of its sides. For example, \( \tan x \) gives the ratio of the opposite side to the adjacent side, while \( \csc x \) provides the reciprocal of \( \sin x \). Understanding these primary functions is essential in simplifying trigonometric expressions.

Trigonometric Functions

Trigonometric functions are mathematical functions of an angle and are fundamental in analyzing periodic phenomena. Each trigonometric function relates to specific ratios of a right triangle's sides.
In trigonometry, these functions are especially important because they represent wave patterns, oscillations, and circular motion in nature.

The six basic trigonometric functions include:

• Sine \( \sin x \) - Ratio of the opposite side to the hypotenuse.
• Cosine \( \cos x \) - Ratio of the adjacent side to the hypotenuse.
• Tangent \( \tan x \) - Ratio of the opposite side to the adjacent side.
• Cosecant \( \csc x \) - Reciprocal of sine.
• Secant \( \sec x \) - Reciprocal of cosine.
• Cotangent \( \cot x \) - Reciprocal of tangent.

In simplifying trigonometric expressions, these functions can be expressed in terms of each other using identities like the Pythagorean identity, angle addition formulas, and reciprocal identities. These identities facilitate the transformation of one function into another, ultimately leading to a simplified form. Mastering these functions and their relationships is crucial in trigonometry.