sin(x - y) sin(x + y) = -tan(y) cos(x + y) + cos(x - y) Use the Addition and Subtraction Formulas, and then simplify. sin(x - y) sin(x + y) cos(x + y) + cos(x - y) (sin(x) cos(y) ) - (sin(x) cos(y) + cos(x) sin(y)) X = (cos(x) cos(y) - sin(x) sin(y)) + (cos(x) cos(y) + sin(x) sin(y)) -2 cos(x) sin(y) 2 cos(x) cos(y) sin (y) X -tan(y) = cos(y)
sin(x - y) sin(x + y) = -tan(y) cos(x + y) + cos(x - y) Use the Addition and Subtraction Formulas, and then simplify. sin(x - y) sin(x + y) cos(x + y) + cos(x - y) (sin(x) cos(y) ) - (sin(x) cos(y) + cos(x) sin(y)) X = (cos(x) cos(y) - sin(x) sin(y)) + (cos(x) cos(y) + sin(x) sin(y)) -2 cos(x) sin(y) 2 cos(x) cos(y) sin (y) X -tan(y) = cos(y)
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
![### Proving the Trigonometric Identity
Given the identity:
\[
\frac{\sin(x - y) - \sin(x + y)}{\cos(x + y) + \cos(x - y)} = -\tan(y)
\]
We will use the Addition and Subtraction Formulas to simplify and verify this identity.
#### Step-by-Step Proof:
1. **Write the Expression Using Addition and Subtraction Formulas:**
\[
\frac{\sin(x - y) - \sin(x + y)}{\cos(x + y) + \cos(x - y)}
\]
Expand using the addition and subtraction formulas:
\[
\sin(x - y) = \sin(x)\cos(y) - \cos(x)\sin(y)
\]
\[
\sin(x + y) = \sin(x)\cos(y) + \cos(x)\sin(y)
\]
2. **Substitute the Expanded Forms:**
\[
\frac{\left(\sin(x)\cos(y) - \cos(x)\sin(y)\right) - \left(\sin(x)\cos(y) + \cos(x)\sin(y)\right)}{\cos(x + y) + \cos(x - y)}
\]
3. **Simplify the Numerator:**
\[
\frac{\left[\sin(x)\cos(y) - \cos(x)\sin(y)\right] - \left[\sin(x)\cos(y) + \cos(x)\sin(y)\right]}{\cos(x + y) + \cos(x - y)}
\]
Distribute the minus sign in the numerator:
\[
\frac{\sin(x)\cos(y) - \cos(x)\sin(y) - \sin(x)\cos(y) - \cos(x)\sin(y)}{\cos(x + y) + \cos(x - y)}
\]
Combine like terms:
\[
\frac{-2\cos(x)\sin(y)}{\cos(x + y) + \cos(x - y)}
\]
4. **Simplify the Denominator Using Addition and Subtraction Formulas:**
\[
\cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y)
\]
\[
\cos(x - y) = \cos(x)\cos(y) + \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6841f1bf-ebcf-44b9-bde6-b1a56299f544%2F75a700e3-56cb-4a0c-ac0b-b8f6598fec56%2F7on0c6u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Proving the Trigonometric Identity
Given the identity:
\[
\frac{\sin(x - y) - \sin(x + y)}{\cos(x + y) + \cos(x - y)} = -\tan(y)
\]
We will use the Addition and Subtraction Formulas to simplify and verify this identity.
#### Step-by-Step Proof:
1. **Write the Expression Using Addition and Subtraction Formulas:**
\[
\frac{\sin(x - y) - \sin(x + y)}{\cos(x + y) + \cos(x - y)}
\]
Expand using the addition and subtraction formulas:
\[
\sin(x - y) = \sin(x)\cos(y) - \cos(x)\sin(y)
\]
\[
\sin(x + y) = \sin(x)\cos(y) + \cos(x)\sin(y)
\]
2. **Substitute the Expanded Forms:**
\[
\frac{\left(\sin(x)\cos(y) - \cos(x)\sin(y)\right) - \left(\sin(x)\cos(y) + \cos(x)\sin(y)\right)}{\cos(x + y) + \cos(x - y)}
\]
3. **Simplify the Numerator:**
\[
\frac{\left[\sin(x)\cos(y) - \cos(x)\sin(y)\right] - \left[\sin(x)\cos(y) + \cos(x)\sin(y)\right]}{\cos(x + y) + \cos(x - y)}
\]
Distribute the minus sign in the numerator:
\[
\frac{\sin(x)\cos(y) - \cos(x)\sin(y) - \sin(x)\cos(y) - \cos(x)\sin(y)}{\cos(x + y) + \cos(x - y)}
\]
Combine like terms:
\[
\frac{-2\cos(x)\sin(y)}{\cos(x + y) + \cos(x - y)}
\]
4. **Simplify the Denominator Using Addition and Subtraction Formulas:**
\[
\cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y)
\]
\[
\cos(x - y) = \cos(x)\cos(y) + \
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