Verify the identity by converting the left side into sines and cosines. (Simplify at each step.) 3 cot(x) sec(x) = 3 csc(x) - 3 sin(x) 3 cot(x) sec(x) = 3 cos(x)/sin(x) 1/(1 3- sin(x) 3 sin(x) sin(x) sin(x)
Verify the identity by converting the left side into sines and cosines. (Simplify at each step.) 3 cot(x) sec(x) = 3 csc(x) - 3 sin(x) 3 cot(x) sec(x) = 3 cos(x)/sin(x) 1/(1 3- sin(x) 3 sin(x) sin(x) sin(x)
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter2: Right Triangle Trigonometry
Section: Chapter Questions
Problem 6GP
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![**Title: Verifying Trigonometric Identities**
**Objective:**
Verify the identity by converting the left side into sines and cosines. Simplify at each step.
**Given Identity:**
\[ \frac{3 \cot(x)}{\sec(x)} = 3 \csc(x) - 3 \sin(x) \]
**Solution:**
1. Rewrite the left side using trigonometric identities:
\[ \frac{3 \cot(x)}{\sec(x)} \]
2. Substitute the identities \(\cot(x) = \frac{\cos(x)}{\sin(x)}\) and \(\sec(x) = \frac{1}{\cos(x)}\):
\[ \frac{3 \cos(x)/\sin(x)}{1/\cos(x)} \]
3. Simplify by multiplying the numerator by the reciprocal of the denominator:
\[ = 3 \left( \frac{\cos(x)}{\sin(x)} \right) \left( \frac{\cos(x)}{1} \right) \]
\[ = 3 (\cos^2(x)/\sin(x)) \]
4. Now, combine under a common denominator:
\[ = \frac{3 \cos^2(x)}{\sin(x)} \]
5. Recognize that \(\cos^2(x) = 1 - \sin^2(x)\) and substitute it:
\[ = \frac{3 (1 - \sin^2(x))}{\sin(x)} \]
\[ = \frac{3 - 3 \sin^2(x)}{\sin(x)} \]
6. Separate the terms in the numerator:
\[ = \frac{3}{\sin(x)} - \frac{3\sin^2(x)}{\sin(x)} \]
7. Simplify each fraction:
\[ = 3 \csc(x) - 3 \sin(x) \]
**Conclusion:**
The left side simplifies to the right side, verifying the identity:
\[ \frac{3 \cot(x)}{\sec(x)} = 3 \csc(x) - 3 \sin(x) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0e38307-1ade-44bc-b712-aaeda4c58098%2Faba4165b-9654-40c6-a3af-7155f711b0b6%2Fypwm1cp_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Verifying Trigonometric Identities**
**Objective:**
Verify the identity by converting the left side into sines and cosines. Simplify at each step.
**Given Identity:**
\[ \frac{3 \cot(x)}{\sec(x)} = 3 \csc(x) - 3 \sin(x) \]
**Solution:**
1. Rewrite the left side using trigonometric identities:
\[ \frac{3 \cot(x)}{\sec(x)} \]
2. Substitute the identities \(\cot(x) = \frac{\cos(x)}{\sin(x)}\) and \(\sec(x) = \frac{1}{\cos(x)}\):
\[ \frac{3 \cos(x)/\sin(x)}{1/\cos(x)} \]
3. Simplify by multiplying the numerator by the reciprocal of the denominator:
\[ = 3 \left( \frac{\cos(x)}{\sin(x)} \right) \left( \frac{\cos(x)}{1} \right) \]
\[ = 3 (\cos^2(x)/\sin(x)) \]
4. Now, combine under a common denominator:
\[ = \frac{3 \cos^2(x)}{\sin(x)} \]
5. Recognize that \(\cos^2(x) = 1 - \sin^2(x)\) and substitute it:
\[ = \frac{3 (1 - \sin^2(x))}{\sin(x)} \]
\[ = \frac{3 - 3 \sin^2(x)}{\sin(x)} \]
6. Separate the terms in the numerator:
\[ = \frac{3}{\sin(x)} - \frac{3\sin^2(x)}{\sin(x)} \]
7. Simplify each fraction:
\[ = 3 \csc(x) - 3 \sin(x) \]
**Conclusion:**
The left side simplifies to the right side, verifying the identity:
\[ \frac{3 \cot(x)}{\sec(x)} = 3 \csc(x) - 3 \sin(x) \]
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