https://file.aiquickdraw.com/mathbot/file/17263234943012egkyqha.jpg

The equation in the image is:

\[
\frac{\cos^2 \frac{\pi}{8} + \cos^2 \frac{3\pi}{8} - \cos^2 \frac{5\pi}{8} + \cos^2 \frac{7\pi}{8}}{8}
\]

### Let's break this down and simplify it step by step:

1. **Cosine Properties**: First, recognize that \( \cos(\theta) \) has symmetry properties:
   - \( \cos\left( \frac{5\pi}{8} \right) = -\cos\left( \frac{3\pi}{8} \right) \)
   - \( \cos\left( \frac{7\pi}{8} \right) = -\cos\left( \frac{\pi}{8} \right) \)

Thus, \( \cos^2 \frac{5\pi}{8} = \cos^2 \frac{3\pi}{8} \) and \( \cos^2 \frac{7\pi}{8} = \cos^2 \frac{\pi}{8} \).

2. **Substitute into the Equation**:
   \[
   \frac{\cos^2 \frac{\pi}{8} + \cos^2 \frac{3\pi}{8} - \cos^2 \frac{3\pi}{8} + \cos^2 \frac{\pi}{8}}{8}
   \]

3. **Simplify**:
   \[
   = \frac{2\cos^2 \frac{\pi}{8}}{8}
   \]

4. **Final Simplification**:
   \[
   = \frac{\cos^2 \frac{\pi}{8}}{4}
   \]

Thus, the simplified expression is \( \frac{\cos^2 \frac{\pi}{8}}{4} \).

Let me know if you'd like further details or have any questions!

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Here are 5 related questions for further practice:
1. How would the expression change if the angles were doubled?
2. What is the value of \( \cos^2 \frac{\pi}{8} \)?
3. How does the symmetry of trigonometric functions help simplify expressions like this?
4. Can the same technique be applied to sine functions in a similar expression?
5. How do you derive identities for \( \cos \left( \frac{n\pi}{8} \right) \) for other values of \(n\)?

**Tip**: When simplifying trigonometric expressions, look for symmetry and periodicity to reduce the complexity of the equation.

Simplify the expression: (cos^2(π/8) + cos^2(3π/8) - cos^2(5π/8) + cos^2(7π/8)) / 8

Learn how to simplify a trigonometric expression involving cosine squares using symmetry properties. This problem covers key trigonometric concepts such as cosine symmetry and basic identities. Suitable for students in Grades 10-12.

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