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Chapter 8: Problem 21

Establish each identity. $$\csc \theta \cdot \cos \theta=\cot \theta$$

Short Answer

Expert verified
The identity is established as \(\frac{1}{\text{sin} \theta} \times \text{cos} \theta = \frac{\text{cos} \theta}{\text{sin} \theta} = \text{cot} \theta\).

Step by step solution

01

Recall the definitions of trigonometric functions

Start by remembering the definitions of the trigonometric functions involved in the given identity. Recall that \(\text{csc} \theta = \frac{1}{\text{sin} \theta}\), \(\text{cos} \theta\), and \(\text{cot} \theta = \frac{\text{cos} \theta}{\text{sin} \theta}\).
02

Substitute the definitions

Replace \(\text{csc} \theta\) and \(\text{cot} \theta\) in the given identity with their definitions: \( \text{csc} \theta \times \text{cos} \theta = \frac{1}{\text{sin} \theta} \times \text{cos} \theta\). The given equation now looks like: \[ \frac{1}{\text{sin} \theta} \times \text{cos} \theta \].
03

Simplify the expression

Next, perform the multiplication in the numerator: \[ \frac{\text{cos} \theta}{\text{sin} \theta} \].
04

Recognize the resulting ratio

Observe that \(\frac{\text{cos} \theta}{\text{sin} \theta} = \text{cot} \theta\), which corresponds to the definition of \(\text{cot} \theta\).

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Key Concepts

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

Cosecant Function
The cosecant function, denoted as \(\text{csc} \theta\), is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function. The formula is simple: \[ \text{csc} \theta = \frac{1}{\text{sin} \theta} \]. This means that wherever sine is not zero, you can compute cosecant by dividing 1 by the sine value. The cosecant function is particularly useful in various trigonometric identities and simplifications, it provides a way to invert the sine function.
Cotangent Function
The cotangent function, represented as \(\text{cot} \theta\), is another essential trigonometric function. It is the reciprocal of the tangent function. The formula to find cotangent is: \[ \text{cot} \theta = \frac{\text{cos} \theta}{\text{sin} \theta} \]. This equation means that you take the cosine of an angle and divide it by the sine of the same angle. Like the sine and cosine functions, cotangent is periodic and can be used in various trigonometric identities and simplifications. Understanding the cotangent function helps in solving many trigonometric equations and identities effectively.
Trigonometric Simplification
Trigonometric simplification involves reducing complex trigonometric expressions into simpler forms. This process makes solving trigonometric identities and equations easier. Take the example from the exercise: \[ \text{csc} \theta \times \text{cos} \theta = \text{cot} \theta \]. The steps involved provide a good illustration of simplification:
  • Recall the definitions.
  • Substitute the identities.
  • Simplify the expression.
  • Recognize the resulting ratio as a known function.
By breaking down the process, complex expressions can be made manageable and more readily solvable.
Trigonometric Definitions
Having a solid understanding of trigonometric definitions is crucial for solving problems and proving identities. The primary definitions you need to remember are:
  • \textbf{Sine (\text{sin})} - Opposite/Hypotenuse
  • \textbf{Cosine (\text{cos})} - Adjacent/Hypotenuse
  • \textbf{Tangent (\text{tan})} - Opposite/Adjacent
  • \textbf{Cosecant (\text{csc})} - \( \frac{1}{\text{sin}} \)
  • \textbf{Secant (\text{sec})} - \( \frac{1}{\text{cos}} \)
  • \textbf{Cotangent (\text{cot})} - \( \frac{\text{cos}}{\text{sin}} \)
These definitions form the backbone of trigonometry. Knowing them thoroughly helps in the simplification and solving of more complex trigonometric expressions and identities.

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