On a separate sheet of paper, prove the following is an identity. cosX/1+sinX + 1+sinX/cosX = 2secX
On a separate sheet of paper, prove the following is an identity. cosX/1+sinX + 1+sinX/cosX = 2secX
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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Question
On a separate sheet of paper, prove the following is an identity.
cosX/1+sinX + 1+sinX/cosX = 2secX
![Certainly! On an educational website, the transcribed content with explanations for the given image would be as follows:
---
**Mathematics Topic: Trigonometric Identities**
**Equation to Verify:**
\[
\frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x} = 2 \sec x
\]
**Explanation and Steps to Verify:**
1. **Expression Simplification:**
The left-hand side of the equation is:
\[
\frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x}
\]
2. **Finding a Common Denominator:**
To combine these two fractions, we need a common denominator. The common denominator for \(1 + \sin x\) and \(\cos x\) is \((1 + \sin x) \cos x \).
3. **Rewriting Each Term:**
Hence, rewrite each fraction with the common denominator:
\[
\frac{\cos x \cdot \cos x}{(1 + \sin x) \cos x} + \frac{(1 + \sin x)(1 + \sin x)}{(1 + \sin x) \cos x}
\]
4. **Simplify the Numerator:**
Simplify the numerators of each fraction:
\[
\frac{\cos^2 x}{(1 + \sin x) \cos x} + \frac{(1 + \sin x)^2}{(1 + \sin x) \cos x}
\]
5. **Combining the Fractions:**
Combine the simplified fractions:
\[
\frac{\cos^2 x + (1 + \sin x)^2}{(1 + \sin x) \cos x}
\]
6. **Expand and Simplify the Numerator:**
Expand \((1 + \sin x)^2\):
\[
(1 + 2\sin x + \sin^2 x)
\]
So the numerator now becomes:
\[
\cos^2 x + 1 + 2\sin x + \sin^2 x
\]
7. **Using Pythagorean Identity:**
Notice that \(\cos^2 x +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe58e15e8-6efe-4a5f-8035-203d9f35ab1f%2Fb0e424a7-1523-4a96-90a9-aa290adadf1d%2Fmcq3u7e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! On an educational website, the transcribed content with explanations for the given image would be as follows:
---
**Mathematics Topic: Trigonometric Identities**
**Equation to Verify:**
\[
\frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x} = 2 \sec x
\]
**Explanation and Steps to Verify:**
1. **Expression Simplification:**
The left-hand side of the equation is:
\[
\frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x}
\]
2. **Finding a Common Denominator:**
To combine these two fractions, we need a common denominator. The common denominator for \(1 + \sin x\) and \(\cos x\) is \((1 + \sin x) \cos x \).
3. **Rewriting Each Term:**
Hence, rewrite each fraction with the common denominator:
\[
\frac{\cos x \cdot \cos x}{(1 + \sin x) \cos x} + \frac{(1 + \sin x)(1 + \sin x)}{(1 + \sin x) \cos x}
\]
4. **Simplify the Numerator:**
Simplify the numerators of each fraction:
\[
\frac{\cos^2 x}{(1 + \sin x) \cos x} + \frac{(1 + \sin x)^2}{(1 + \sin x) \cos x}
\]
5. **Combining the Fractions:**
Combine the simplified fractions:
\[
\frac{\cos^2 x + (1 + \sin x)^2}{(1 + \sin x) \cos x}
\]
6. **Expand and Simplify the Numerator:**
Expand \((1 + \sin x)^2\):
\[
(1 + 2\sin x + \sin^2 x)
\]
So the numerator now becomes:
\[
\cos^2 x + 1 + 2\sin x + \sin^2 x
\]
7. **Using Pythagorean Identity:**
Notice that \(\cos^2 x +
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