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Chapter 2: Problem 29

\(\frac{\tan x \cot x}{\cos x}=\sec x\)

Short Answer

Expert verified
Yes, the simplification of the equation shows that the left hand side is equal to the right hand side which proves the trigonometric identity \(\frac{\tan x \cot x}{\cos x}=\sec x\).

Step by step solution

01

Simplify the left hand side

Replace \(\tan x\) and \(\cot x\) with their definitions in terms of \(\sin x\) and \(\cos x\). So, \(\tan x = \frac{\sin x}{\cos x}\) and \(\cot x = \frac{\cos x}{\sin x}\). Substituting these into the equation gives \(\frac{\frac{\sin x}{\cos x} \cdot \frac{\cos x}{\sin x}}{\cos x}\).
02

Simplify the denominator

The \(\sin x\) and \(\cos x\) cancel out in the numerator, leaving just 1. The denominator is \(\cos x\), so we get \(\frac{1}{\cos x}\).
03

Identify the resulting trigonometric identity

The expression \(\frac{1}{\cos x}\) is the definition of \(\sec x\), so the left hand side equals the right hand side, thus proving the identity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Trigonometric Functions
Trigonometric functions are fundamental in mathematics, relating the angles of triangles to their side lengths. There are six primary trigonometric functions: sine (\(\sin x\)), cosine (\(\cos x\)), tangent (\(\tan x\)), cotangent (\(\cot x\)), secant (\(\sec x\)), and cosecant (\(\csc x\)). These functions are essential in various scientific and engineering fields. For example:
  • \(\tan x\) is defined as \(\frac{\sin x}{\cos x}\)
  • \(\cot x\) equals \(\frac{\cos x}{\sin x}\)
  • \(\sec x\) is \(\frac{1}{\cos x}\)
Each function helps to connect the geometric world with algebra, allowing us to model and solve real-world problems. Understanding these relationships aids in simplifying and proving complex trigonometric identities.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions involves reducing them to more manageable forms. This process often requires substituting equivalent trigonometric functions and canceling out terms. Let's see how this applies to our exercise:We began with \(\frac{\tan x \cot x}{\cos x}\). By substituting the definitions:
  • \(\tan x = \frac{\sin x}{\cos x}\)
  • \(\cot x = \frac{\cos x}{\sin x}\)
We get \(\frac{\frac{\sin x}{\cos x} \cdot \frac{\cos x}{\sin x}}{\cos x}\). Here, \(\sin x\) and \(\cos x\) cancel each other out, simplifying the expression to \(\frac{1}{\cos x}\). Recognizing how expressions can be simplified is a key step in solving trigonometric identities and equations.
Techniques for Proving Trigonometric Identities
Proving trigonometric identities involves showing that one side of an equation can be transformed into the other using known identities. It's like solving a puzzle where each piece represents a small trigonometric truth. To illustrate this:Starting with \(\frac{\tan x \cot x}{\cos x} = \sec x\), we needed to show that both sides could be expressed identically:
  • First, use identity definitions to rewrite expressions.
  • Simplify by canceling terms as seen with the numerator \(\frac{\sin x}{\cos x} \cdot \frac{\cos x}{\sin x} = 1\)
  • The remaining expression \(\frac{1}{\cos x}\) directly translates to \(\sec x\)
Proving identities requires creativity and knowledge of trigonometric identities, ensuring one side matches the other exactly. Mastering this skill builds a deep understanding of mathematical relationships and promotes critical thinking.

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