Problem 20 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Understand the Trigonometric Functions

The expression we need to simplify is \( \tan x \cos x \csc x \). Let's identify the trigonometric functions involved: \( \tan x = \frac{\sin x}{\cos x} \), \( \cos x \) is just the cosine function, and \( \csc x = \frac{1}{\sin x} \), which is the reciprocal of the sine function.

Step 2: Substitute Trigonometric Identities

Substitute the identities into the expression: \( \tan x = \frac{\sin x}{\cos x} \) and \( \csc x = \frac{1}{\sin x} \). Plugging these into the expression gives: \[ \left( \frac{\sin x}{\cos x} \right) \cdot \cos x \cdot \left( \frac{1}{\sin x} \right) \].

Step 3: Simplify the Expression

Simplify the expression by cancelling out terms in the numerator and denominator. The \( \sin x \) in the numerator of \( \tan x \) cancels out with \( \sin x \) in the denominator of \( \csc x \), and \( \cos x \) in the numerator and denominator also cancels: \[ \frac{\sin x \cdot \cos x}{\cos x \cdot \sin x} = 1 \].

Step 4: Present the Final Simplified Form

After cancelling out the terms, we are left with 1. Therefore, the simplified form of the original trigonometric expression is 1.

Key Concepts

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves making them easier to work with by using mathematical identities and properties to reduce their complexity. In our example, let's break down the expression \( \tan x \cos x \csc x \). First, identify each component by looking at their basic definitions. Recognize that we are dealing with trigonometric functions like tangent, cosine, and cosecant.
Understanding these components, the next step is to substitute known identities. This helps in converting the expression into a form that's easier to simplify. For instance, by expressing \( \tan x \) as \( \frac{\sin x}{\cos x} \) and \( \csc x \) as \( \frac{1}{\sin x} \), we exploit the relationships between the trigonometric functions to find common terms that cancel each other out.
By doing this, you can streamline the expression and eventually, reduce it to its simplest form. Simplification not only provides elegance but also makes further calculations and applications of the expression easier. In this case, identifying and canceling terms results in the expression simplifying to 1.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in geometry and calculus. They describe relationships between the angles and sides of right-angled triangles. Key trigonometric functions include sine (\(\sin x\)), cosine (\(\cos x\)), and tangent (\(\tan x\)).

• **Sine (\(\sin x\))** describes the ratio of the opposite side to the hypotenuse.
• **Cosine (\(\cos x\))** describes the ratio of the adjacent side to the hypotenuse.
• **Tangent (\(\tan x\))** describes the ratio of the opposite side to the adjacent side, which can also be represented as \(\tan x = \frac{\sin x}{\cos x}\).

Comprehending these functions is pivotal because they form the base that allows us to express more complex mathematical relationships. For example, in simplifying the trigonometric expression, recognizing how these functions interrelate allows for systematic reduction of equations, leveraging identities such as the reciprocal and quotient identities.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the basic trigonometric functions through the principle of reciprocity. Three main reciprocal functions are cosecant (\(\csc x\)), secant (\(\sec x\)), and cotangent (\(\cot x\)). They are useful for solving problems involving trigonometric ratios and are inversely related to the primary trigonometric functions.

• **Cosecant (\(\csc x\))** is the reciprocal of sine: \(\csc x = \frac{1}{\sin x}\).
• **Secant (\(\sec x\))** is the reciprocal of cosine: \(\sec x = \frac{1}{\cos x}\).
• **Cotangent (\(\cot x\))** is the reciprocal of tangent: \(\cot x = \frac{1}{\tan x}\) or \(\cot x = \frac{\cos x}{\sin x}\).

Understanding reciprocal functions is key to transforming trigonometric expressions. In simplifying \( \tan x \cos x \csc x \), reciprocal identities allow us to replace complex fractions with simpler forms, paving the way for easier manipulation. By substituting these functions appropriately, we clear paths to reduce expressions and arrive at straightforward solutions.