Simplifying Trigonometric Expressions Using Fundamental Identities

Q1. Write the expression $\sec^2(-\theta) - \cos^2(-\theta) - \tan^2(-\theta)$ in terms of sine and cosine, and then simplify so that no quotients appear in the final expression. All functions should be of $\theta$ only.

Background

Topic: Trigonometric Identities and Simplification

This question tests your understanding of fundamental trigonometric identities, including reciprocal, quotient, and even-odd identities. You are asked to rewrite and simplify a trigonometric expression so that it only involves sine and cosine functions, with no quotients.

Key Terms and Formulas

• Reciprocal identities:

  • $\sec\theta = \frac{1}{\cos\theta}$

  • $\tan\theta = \frac{\sin\theta}{\cos\theta}$

• Even-odd identities:

  • $\cos(-\theta) = \cos\theta$

  • $\sin(-\theta) = -\sin\theta$

  • $\tan(-\theta) = -\tan\theta$

  • $\sec(-\theta) = \sec\theta$

• Pythagorean identity:

  • $\tan^2\theta + 1 = \sec^2\theta$

Step-by-Step Guidance

1. Apply the even-odd identities to rewrite all functions in terms of $\theta$ (not $-\theta$): $\sec^2(-\theta) = \sec^2\theta$, $\cos^2(-\theta) = \cos^2\theta$, $\tan^2(-\theta) = \tan^2\theta$.

   

2. Rewrite each function in terms of sine and cosine using reciprocal and quotient identities: $\sec^2\theta = \left(\frac{1}{\cos\theta}\right)^2 = \frac{1}{\cos^2\theta}$ $\tan^2\theta = \left(\frac{\sin\theta}{\cos\theta}\right)^2 = \frac{\sin^2\theta}{\cos^2\theta}$

3. Substitute these expressions into the original equation: $\frac{1}{\cos^2\theta} - \cos^2\theta - \frac{\sin^2\theta}{\cos^2\theta}$

4. Combine terms over a common denominator to eliminate quotients: $\frac{1 - \sin^2\theta}{\cos^2\theta} - \cos^2\theta$

5. Recall the Pythagorean identity $1 - \sin^2\theta = \cos^2\theta$ and substitute it in to further simplify.

Try solving on your own before revealing the answer!

Final Answer: $0$

After substituting and simplifying, the expression reduces to $0$because$\cos^2\theta - \cos^2\theta = 0$.

This demonstrates the power of fundamental identities in simplifying trigonometric expressions.