21 Prove the identity.cos x (1 ²x) 2 2. - sin sec X,
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![## Prove the Identity
Prove the identity:
\[ \cos^2 x \left( 1 - \sec^2 x \right) = -\sin^2 x \]
### Explanation
1. **Start with the left side:**
\[ \cos^2 x (1 - \sec^2 x) \]
2. **Use the identity for secant:**
\[ \sec^2 x = \frac{1}{\cos^2 x} \]
3. **Substitute into the equation:**
\[ \cos^2 x \left( 1 - \frac{1}{\cos^2 x} \right) \]
4. **Simplify:**
\[ \cos^2 x \left( \frac{\cos^2 x - 1}{\cos^2 x} \right) = \cos^2 x \cdot \left( -\frac{\sin^2 x}{\cos^2 x} \right) \]
5. **Cancel \(\cos^2 x\):**
\[ -\sin^2 x \]
Thus, the identity is proved:
\[ \cos^2 x (1 - \sec^2 x) = -\sin^2 x \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6c772740-f68c-4402-a39b-79004599ac50%2F80cf7a95-9790-4310-9d69-86d5acf2187f%2F1j11ieb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Prove the Identity
Prove the identity:
\[ \cos^2 x \left( 1 - \sec^2 x \right) = -\sin^2 x \]
### Explanation
1. **Start with the left side:**
\[ \cos^2 x (1 - \sec^2 x) \]
2. **Use the identity for secant:**
\[ \sec^2 x = \frac{1}{\cos^2 x} \]
3. **Substitute into the equation:**
\[ \cos^2 x \left( 1 - \frac{1}{\cos^2 x} \right) \]
4. **Simplify:**
\[ \cos^2 x \left( \frac{\cos^2 x - 1}{\cos^2 x} \right) = \cos^2 x \cdot \left( -\frac{\sin^2 x}{\cos^2 x} \right) \]
5. **Cancel \(\cos^2 x\):**
\[ -\sin^2 x \]
Thus, the identity is proved:
\[ \cos^2 x (1 - \sec^2 x) = -\sin^2 x \]
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