Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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10. no calculater way please
![**Trigonometric Identity Verification**
Show that \( \tan(x) + \cot(x) = \sec(x) \csc(x) \). (Verify this identity.)
**Verification:**
To verify the trigonometric identity, we will transform both sides of the equation to make them equal.
1. **Left Side:**
\[ \tan(x) + \cot(x) \]
We know that:
\[ \tan(x) = \frac{\sin(x)}{\cos(x)} \]
\[ \cot(x) = \frac{\cos(x)}{\sin(x)} \]
Combining these, we get:
\[ \tan(x) + \cot(x) = \frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)} \]
To add these fractions, find a common denominator:
\[ \tan(x) + \cot(x) = \frac{\sin^2(x) + \cos^2(x)}{\sin(x)\cos(x)} \]
Using the Pythagorean identity \( \sin^2(x) + \cos^2(x) = 1 \):
\[ \tan(x) + \cot(x) = \frac{1}{\sin(x)\cos(x)} \]
2. **Right Side:**
\[ \sec(x) \csc(x) \]
We know that:
\[ \sec(x) = \frac{1}{\cos(x)} \]
\[ \csc(x) = \frac{1}{\sin(x)} \]
Multiplying these, we get:
\[ \sec(x) \csc(x) = \frac{1}{\cos(x)} \cdot \frac{1}{\sin(x)} = \frac{1}{\sin(x)\cos(x)} \]
3. **Conclusion:**
Since both the left side and the right side simplify to:
\[ \frac{1}{\sin(x)\cos(x)} \]
the identity is verified. Therefore,
\[ \tan(x) + \cot(x) = \sec(x) \csc(x) \]
is true.](https://content.bartleby.com/qna-images/question/4d6261b1-d641-4c15-8b49-7d4d38863372/30378dc5-4f56-47c8-a54a-f861312ad167/nc271g_processed.png)
Transcribed Image Text:**Trigonometric Identity Verification**
Show that \( \tan(x) + \cot(x) = \sec(x) \csc(x) \). (Verify this identity.)
**Verification:**
To verify the trigonometric identity, we will transform both sides of the equation to make them equal.
1. **Left Side:**
\[ \tan(x) + \cot(x) \]
We know that:
\[ \tan(x) = \frac{\sin(x)}{\cos(x)} \]
\[ \cot(x) = \frac{\cos(x)}{\sin(x)} \]
Combining these, we get:
\[ \tan(x) + \cot(x) = \frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)} \]
To add these fractions, find a common denominator:
\[ \tan(x) + \cot(x) = \frac{\sin^2(x) + \cos^2(x)}{\sin(x)\cos(x)} \]
Using the Pythagorean identity \( \sin^2(x) + \cos^2(x) = 1 \):
\[ \tan(x) + \cot(x) = \frac{1}{\sin(x)\cos(x)} \]
2. **Right Side:**
\[ \sec(x) \csc(x) \]
We know that:
\[ \sec(x) = \frac{1}{\cos(x)} \]
\[ \csc(x) = \frac{1}{\sin(x)} \]
Multiplying these, we get:
\[ \sec(x) \csc(x) = \frac{1}{\cos(x)} \cdot \frac{1}{\sin(x)} = \frac{1}{\sin(x)\cos(x)} \]
3. **Conclusion:**
Since both the left side and the right side simplify to:
\[ \frac{1}{\sin(x)\cos(x)} \]
the identity is verified. Therefore,
\[ \tan(x) + \cot(x) = \sec(x) \csc(x) \]
is true.
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