Verify the identity. (Simplify at each step.) tan t cot t = 1 tan t cot t = cos t cos t sin t = 1
Verify the identity. (Simplify at each step.) tan t cot t = 1 tan t cot t = cos t cos t sin t = 1
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
Related questions
Question
![### Verifying Trigonometric Identity
**Objective:**
Verify the trigonometric identity \( \tan t \cot t = 1 \). (Simplify at each step.)
**Step-by-Step Simplification:**
1. We start with the given identity:
\[
\tan t \cot t = 1
\]
2. Express \(\cot t\) (cotangent of \( t \)) in terms of \(\tan t\) (tangent of \( t \)):
\[
\cot t = \frac{1}{\tan t}
\]
3. Substitute \(\frac{1}{\tan t}\) for \(\cot t\):
\[
\tan t \cdot \frac{1}{\tan t} = 1
\]
4. This simplifies to:
\[
1 = 1
\]
The identity holds true as the left-hand side simplifies to the right-hand side.
**Additional Explanation:**
In the diagram provided, the verification process shows the following:
- Left-hand side: \( \tan t \cot t \)
- Substituting the definition of \(\cot t\): \( \tan t \cdot \frac{1}{\tan t} \)
- Simplifying the multiplication: \( \frac{\tan t}{\tan t} = 1 \)
Therefore, \( \tan t \cot t = 1 \), confirming the identity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0e38307-1ade-44bc-b712-aaeda4c58098%2F2ef83190-68d2-4a8f-90b8-af5b565bd9d8%2Fsxo3soi_processed.png&w=3840&q=75)
Transcribed Image Text:### Verifying Trigonometric Identity
**Objective:**
Verify the trigonometric identity \( \tan t \cot t = 1 \). (Simplify at each step.)
**Step-by-Step Simplification:**
1. We start with the given identity:
\[
\tan t \cot t = 1
\]
2. Express \(\cot t\) (cotangent of \( t \)) in terms of \(\tan t\) (tangent of \( t \)):
\[
\cot t = \frac{1}{\tan t}
\]
3. Substitute \(\frac{1}{\tan t}\) for \(\cot t\):
\[
\tan t \cdot \frac{1}{\tan t} = 1
\]
4. This simplifies to:
\[
1 = 1
\]
The identity holds true as the left-hand side simplifies to the right-hand side.
**Additional Explanation:**
In the diagram provided, the verification process shows the following:
- Left-hand side: \( \tan t \cot t \)
- Substituting the definition of \(\cot t\): \( \tan t \cdot \frac{1}{\tan t} \)
- Simplifying the multiplication: \( \frac{\tan t}{\tan t} = 1 \)
Therefore, \( \tan t \cot t = 1 \), confirming the identity.
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