Problem 26 - Contemporary Precalculus

Question

State whether or not the equation is an identity. If it is an identity, prove it. $$\cot x=\frac{\csc x}{\sec x}$$

Step-by-Step Solution

Answer

$$ Solution: After using basic trigonometric identities to convert the given equation into simpler terms, we find that the equation simplifies to: $$\frac{\cos x}{\sin x} = \frac{\cos x}{\sin x}.$$ Since both sides of the equation are equal, we can conclude that the given equation is an identity.

Step 1: Convert into Basic Trigonometric Functions

We'll start by converting cotangent, cosecant, and secant into basic trigonometric functions – sine, cosine, and tangent. Recall the following relationships: $$\cot x = \frac{1}{\tan x}$$ $$\csc x = \frac{1}{\sin x}$$ $$\sec x = \frac{1}{\cos x}$$ Using these relationships, we can rewrite the given equation as: $$\frac{1}{\tan x}=\frac{\frac{1}{\sin x}}{\frac{1}{\cos x}}$$

Step 2: Simplify the Equation

Now, simplify the equation by working on the right side: $$\frac{1}{\tan x}=\frac{1}{\sin x} \cdot \frac{\cos x}{1}$$

Step 3: Use the Tangent Identity

Recall the tangent identity: $$\tan x = \frac{\sin x}{\cos x}$$ Using this identity, rewrite the equation: $$\frac{1}{\frac{\sin x}{\cos x}}=\frac{\cos x}{\sin x}$$

Step 4: Simplify the Left Side of the Equation

We can further simplify the left side by multiplying both the numerator and the denominator by \(\cos{x}\): $$\frac{\cos x}{\sin x}=\frac{\cos x}{\sin x}$$ Now both sides of the equation are the same, proving that the given equation is an identity: $$\cot x=\frac{\csc x}{\sec x}$$

Key Concepts

Cotangent

The cotangent function is a lesser-known member of the trigonometric family that pairs closely with tangent. It is often represented by \( \cot x \) and defined as the reciprocal of the tangent function. In simpler terms, this means:

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

Cotangent plays an important role in trigonometry when solving problems involving right triangles and periodic functions. To understand how it fits into the larger picture, remember its close relationship with tangent. While tangent is the ratio of the sine to cosine, cotangent flips this relationship around. That is, it emphasizes how one can approach problems differently by simply using reciprocal identities. As seen in proving identities, recognizing these reciprocal relationships helps simplify and solve complex equations.

Cosecant

The cosecant function is the reciprocal of the sine function and is represented by \( \csc x \). Cosecant is less common in basic trigonometry but becomes essential when exploring deeper mathematical concepts. The formula for cosecant is:

• \( \csc x = \frac{1}{\sin x} \)

Understanding cosecant helps in cases where it's easier or more convenient to use its reciprocal function rather than dealing directly with sine. For example, when dealing with identities or simplifying expressions, changing sine to cosecant can lead to more straightforward calculations. As we see in this exercise, \( \csc x \) is used in the identity \( \cot x = \frac{\csc x}{\sec x} \). Recognizing the reciprocal relationship here helps to manipulate the trigonometric functions and solve or prove identities efficiently.

Secant

The secant function is the reciprocal of the cosine function and is denoted by \( \sec x \). Like cosecant, secant is not commonly used in basic trigonometry but proves valuable in various mathematical applications, especially in calculus and advanced trigonometry. The basic definition of secant is:

• \( \sec x = \frac{1}{\cos x} \)

Secant is useful in expressions where simplifying or proving identities requires flipping the cosine function. It is part of the relationship that binds the identity \( \cot x = \frac{\csc x}{\sec x} \). Recognizing the role of \( \sec x \) in such identities allows mathematicians to reframe and simplify complex expressions. Knowing how to effectively use secant helps in solving problems that might be cumbersome with just sine and cosine.