Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Fundamentals of Trigonometric Identities
Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.
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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: \[ \cos^2 x (1 - \sec^2 x) = -\sin^2 x \]
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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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Consider the given identity cos2x1-sec2x=-sin2x.

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