Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![## Mathematical Equation Explanation
The equation shown in the image is:
\[
\frac{1}{\sin t - 1} + \frac{1}{\sin t + 1} = -2 \tan t \, \sec t
\]
### Breakdown of the Equation:
1. **Left Side:**
- \(\frac{1}{\sin t - 1}\): This term represents the reciprocal of the expression \((\sin t - 1)\).
- \(\frac{1}{\sin t + 1}\): This term represents the reciprocal of the expression \((\sin t + 1)\).
2. **Right Side:**
- \(-2 \tan t \, \sec t\): The right side is a trigonometric expression involving the tangent and secant functions of \(t\), multiplied by \(-2\).
### Concepts Involved:
- **Sine (\(\sin\))**: A primary trigonometric function representing the ratio of the opposite side to the hypotenuse in a right triangle.
- **Tangent (\(\tan\))**: A trigonometric function representing the ratio of the sine to the cosine of an angle.
- **Secant (\(\sec\))**: The reciprocal of the cosine function.
### Purpose:
This equation may be part of a problem involving trigonometric identities or transformations. Understanding how to manipulate and simplify such equations is essential in trigonometry.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4116597-78e6-477d-91b2-9fa4cad4d0f7%2F7be42d38-2a55-4a15-8b05-ae680a65d40d%2F7dmv7vr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Mathematical Equation Explanation
The equation shown in the image is:
\[
\frac{1}{\sin t - 1} + \frac{1}{\sin t + 1} = -2 \tan t \, \sec t
\]
### Breakdown of the Equation:
1. **Left Side:**
- \(\frac{1}{\sin t - 1}\): This term represents the reciprocal of the expression \((\sin t - 1)\).
- \(\frac{1}{\sin t + 1}\): This term represents the reciprocal of the expression \((\sin t + 1)\).
2. **Right Side:**
- \(-2 \tan t \, \sec t\): The right side is a trigonometric expression involving the tangent and secant functions of \(t\), multiplied by \(-2\).
### Concepts Involved:
- **Sine (\(\sin\))**: A primary trigonometric function representing the ratio of the opposite side to the hypotenuse in a right triangle.
- **Tangent (\(\tan\))**: A trigonometric function representing the ratio of the sine to the cosine of an angle.
- **Secant (\(\sec\))**: The reciprocal of the cosine function.
### Purpose:
This equation may be part of a problem involving trigonometric identities or transformations. Understanding how to manipulate and simplify such equations is essential in trigonometry.
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