sin' t – cos' t Prove tan?t – 1 cos? t
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
Prove
sin4t-cos4t/cos2t = tan2t - 1
![**Trigonometric Identity Proof**
In this exercise, we are tasked with proving the following trigonometric identity:
\[
\frac{\sin^4 t - \cos^4 t}{\cos^2 t} = \tan^2 t - 1
\]
### Steps to Prove:
1. **Start with the Left-Hand Side (LHS):**
\[
\frac{\sin^4 t - \cos^4 t}{\cos^2 t}
\]
2. **Use the algebraic identity for difference of squares:**
\[
\sin^4 t - \cos^4 t = (\sin^2 t - \cos^2 t)(\sin^2 t + \cos^2 t)
\]
3. **Recall the Pythagorean identity:**
\[
\sin^2 t + \cos^2 t = 1
\]
Therefore, substitute back in:
\[
\frac{(\sin^2 t - \cos^2 t)(1)}{\cos^2 t} = \frac{\sin^2 t - \cos^2 t}{\cos^2 t}
\]
4. **Separate the fraction:**
\[
\frac{\sin^2 t}{\cos^2 t} - \frac{\cos^2 t}{\cos^2 t} = \tan^2 t - 1
\]
5. **Conclusion:**
Both sides simplify to \(\tan^2 t - 1\), hence proving the identity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0fd4670-9e52-488d-82d1-dba81b363ab0%2F8d501c0c-4fef-475f-b200-e90bf065cc81%2F6ln2v4j_processed.png&w=3840&q=75)
Transcribed Image Text:**Trigonometric Identity Proof**
In this exercise, we are tasked with proving the following trigonometric identity:
\[
\frac{\sin^4 t - \cos^4 t}{\cos^2 t} = \tan^2 t - 1
\]
### Steps to Prove:
1. **Start with the Left-Hand Side (LHS):**
\[
\frac{\sin^4 t - \cos^4 t}{\cos^2 t}
\]
2. **Use the algebraic identity for difference of squares:**
\[
\sin^4 t - \cos^4 t = (\sin^2 t - \cos^2 t)(\sin^2 t + \cos^2 t)
\]
3. **Recall the Pythagorean identity:**
\[
\sin^2 t + \cos^2 t = 1
\]
Therefore, substitute back in:
\[
\frac{(\sin^2 t - \cos^2 t)(1)}{\cos^2 t} = \frac{\sin^2 t - \cos^2 t}{\cos^2 t}
\]
4. **Separate the fraction:**
\[
\frac{\sin^2 t}{\cos^2 t} - \frac{\cos^2 t}{\cos^2 t} = \tan^2 t - 1
\]
5. **Conclusion:**
Both sides simplify to \(\tan^2 t - 1\), hence proving the identity.
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