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Chapter 20: Problem 20

Prove the given identities. $$\frac{\csc \theta}{\sec \theta}=\cot \theta$$

Short Answer

Expert verified
The identity is proven as \( \frac{\csc \theta}{\sec \theta} = \cot \theta \).

Step by step solution

01

Convert to Sin and Cos

We begin by rewriting the trigonometric functions \( \csc \theta \) and \( \sec \theta \) in terms of \( \sin \theta \) and \( \cos \theta \). We know \( \csc \theta = \frac{1}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \). Substitute these into the left side of the equation: \\[ \frac{\csc \theta}{\sec \theta} = \frac{\frac{1}{\sin \theta}}{\frac{1}{\cos \theta}} \]
02

Simplify the Fraction

Simplify the complex fraction by multiplying by the reciprocal: \\[ \frac{\frac{1}{\sin \theta}}{\frac{1}{\cos \theta}} = \frac{1}{\sin \theta} \times \frac{\cos \theta}{1} = \frac{\cos \theta}{\sin \theta} \]
03

Recognize Cotangent Function

The expression \ \frac{\cos \theta}{\sin \theta} \ is the definition of the \ \cot \theta \ function. Therefore, we can simplify further: \\[ \frac{\cos \theta}{\sin \theta} = \cot \theta \]

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Key Concepts

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

Cosecant
The cosecant function, often abbreviated as \( \csc \theta \), is one of the six fundamental trigonometric functions. It's considered the reciprocal of the sine function.
Understanding the relation between trigonometric functions and their reciprocals is key to tackling identity proofs like the one in this exercise.
  • The sine function is defined as \( \sin \theta = \frac{{\text{opposite}}}{\text{hypotenuse}} \) in a right triangle.
  • Cosecant, therefore, is defined as the reciprocal: \( \csc \theta = \frac{1}{\sin \theta} = \frac{{\text{hypotenuse}}}{\text{opposite}} \).
The role of the cosecant is crucial because it transforms a division involving \(\sin \theta \) into multiplication, easing the process of simplifying identities. Here, replacing \( \csc \theta \) allows us to transition from a complex fraction to a simple trigonometric form, facilitating the discovery of other identities.
Secant
The secant function, denoted as \(\sec \theta \), complements the cosecant as its counterpart with respect to the cosine function. Just like the other trigonometric functions, secant is based on a fundamental relationship:
  • The cosine function is described by \(\cos \theta = \frac{{\text{adjacent}}}{\text{hypotenuse}}\) in a right triangle.
  • As a reciprocal, secant is defined by \(\sec \theta = \frac{1}{\cos \theta} = \frac{{\text{hypotenuse}}}{\text{adjacent}}\).
Secant appears less frequently in calculations than sine or cosine, but it is essential in identities. Like in our exercise, converting \(\sec \theta\) helps to simplify and solve the expression. By transforming \(\sec \theta\), the identity can be expressed solely in terms of sine and cosine, making it easier to manipulate and simplify.
Cotangent
Cotangent, denoted as \(\cot \theta \), is a vital trigonometric function that often appears in both identity proofs and calculus. It can be expressed in a couple of ways, showcasing its versatility:
  • Cotan is the reciprocal of tangent: \(\cot \theta = \frac{1}{\tan \theta}\).
  • Using sine and cosine, we have \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).
This second form, \(\frac{\cos \theta}{\sin \theta}\), highlights the relationship between the functions involved in identity proofs. In our exercise, recognizing \(\frac{\cos \theta}{\sin \theta}\) as \(\cot \theta\) completes the proof, demonstrating how reciprocal relationships simplify expressions and lead to the solutions. Understanding these connections makes it easier to grasp and solve complex trigonometric identities.

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Most popular questions from this chapter

$$\text { Find the exact value of } \cos 2 x+\sin 2 x \tan x$$.

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