4. Add the two fractions and simplify the result: 5. Verify that the expression is an identity: sin ²0 cos 8 COS X secx sin x + CSC X sec8 - cos 8 DEL

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°.
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### Trigonometry Problems and Identities

#### 4. Add the two fractions and simplify the result:
\[ \frac{\cos x}{\sec x} + \frac{\sin x}{\csc x} \]

**Solution:**
To simplify the given expression, we follow these steps:

1. Recall that \(\sec x\) is the reciprocal of \(\cos x\) and \(\csc x\) is the reciprocal of \(\sin x\). Therefore:
   \[ \sec x = \frac{1}{\cos x} \quad \text{and} \quad \csc x = \frac{1}{\sin x} \]
2. Substitute these values into the fractions:
   \[ \frac{\cos x}{\sec x} = \frac{\cos x}{\frac{1}{\cos x}} = \cos^2 x \]
   \[ \frac{\sin x}{\csc x} = \frac{\sin x}{\frac{1}{\sin x}} = \sin^2 x \]
3. Add the two simplified fractions:
   \[ \cos^2 x + \sin^2 x \]
4. Recall the Pythagorean identity for sine and cosine:
   \[ \cos^2 x + \sin^2 x = 1 \]

Therefore, the simplified expression is:
\[ \boxed{1} \]

#### 5. Verify that the expression is an identity:
\[ \frac{\sin^2 \theta}{\cos \theta} = \sec \theta - \cos \theta \]

**Verification:**
To verify this identity, we perform the following steps:

1. Express \(\sec \theta\) in terms of \(\cos \theta\):
   \[ \sec \theta = \frac{1}{\cos \theta} \]
2. Substitute this value into the right side of the equation:
   \[ \sec \theta - \cos \theta = \frac{1}{\cos \theta} - \cos \theta \]
3. Combine the terms over a common denominator:
   \[ \frac{1}{\cos \theta} - \cos \theta = \frac{1 - \cos^2 \theta}{\cos \theta} \]
4. Use the Pythagorean identity \(\sin^2 \theta
Transcribed Image Text:### Trigonometry Problems and Identities #### 4. Add the two fractions and simplify the result: \[ \frac{\cos x}{\sec x} + \frac{\sin x}{\csc x} \] **Solution:** To simplify the given expression, we follow these steps: 1. Recall that \(\sec x\) is the reciprocal of \(\cos x\) and \(\csc x\) is the reciprocal of \(\sin x\). Therefore: \[ \sec x = \frac{1}{\cos x} \quad \text{and} \quad \csc x = \frac{1}{\sin x} \] 2. Substitute these values into the fractions: \[ \frac{\cos x}{\sec x} = \frac{\cos x}{\frac{1}{\cos x}} = \cos^2 x \] \[ \frac{\sin x}{\csc x} = \frac{\sin x}{\frac{1}{\sin x}} = \sin^2 x \] 3. Add the two simplified fractions: \[ \cos^2 x + \sin^2 x \] 4. Recall the Pythagorean identity for sine and cosine: \[ \cos^2 x + \sin^2 x = 1 \] Therefore, the simplified expression is: \[ \boxed{1} \] #### 5. Verify that the expression is an identity: \[ \frac{\sin^2 \theta}{\cos \theta} = \sec \theta - \cos \theta \] **Verification:** To verify this identity, we perform the following steps: 1. Express \(\sec \theta\) in terms of \(\cos \theta\): \[ \sec \theta = \frac{1}{\cos \theta} \] 2. Substitute this value into the right side of the equation: \[ \sec \theta - \cos \theta = \frac{1}{\cos \theta} - \cos \theta \] 3. Combine the terms over a common denominator: \[ \frac{1}{\cos \theta} - \cos \theta = \frac{1 - \cos^2 \theta}{\cos \theta} \] 4. Use the Pythagorean identity \(\sin^2 \theta
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