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Let's solve each part step by step.

Given:
\[
\theta = \frac{3\pi}{4}
\]

1. **(a) \( \cos(-\theta) \)**

   Using the even property of the cosine function, \( \cos(-\theta) = \cos(\theta) \). So:
   \[
   \cos\left(-\frac{3\pi}{4}\right) = \cos\left(\frac{3\pi}{4}\right)
   \]
   The value of \( \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \).
   
   Therefore, \( \cos(-\theta) = -\frac{\sqrt{2}}{2} \).

2. **(b) \( \cos^2 \theta \)**

   \( \cos^2 \theta \) means \( (\cos \theta)^2 \).
   \[
   \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}
   \]
   So,
   \[
   \cos^2 \theta = \left(-\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} = \frac{1}{2}
   \]

3. **(c) \( \cos 2\theta \)**

   Using the double-angle formula, \( \cos 2\theta = 2\cos^2 \theta - 1 \).
   We already found \( \cos^2 \theta = \frac{1}{2} \), so:
   \[
   \cos 2\theta = 2 \times \frac{1}{2} - 1 = 1 - 1 = 0
   \]

### Final Answers:
- (a) \( \cos(-\theta) = -\frac{\sqrt{2}}{2} \)
- (b) \( \cos^2 \theta = \frac{1}{2} \)
- (c) \( \cos 2\theta = 0 \)

Would you like a more detailed breakdown of each step?

Here are 5 related questions for practice:
1. What is \( \sin(-\theta) \) if \( \theta = \frac{3\pi}{4} \)?
2. Calculate \( \tan(\theta) \) for \( \theta = \frac{3\pi}{4} \).
3. Use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to verify \( \cos^2 \theta \).
4. How does the value of \( \cos(2\theta) \) change if \( \theta = \frac{\pi}{3} \) instead?
5. What is the value of \( \sec(\theta) \) if \( \theta = \frac{3\pi}{4} \)?

**Tip:** For angles like \( \frac{3\pi}{4} \), remember they fall in the second quadrant, where cosine values are negative.

If θ = 3π/4, find the exact value of each expression below.
(a) cos(-θ)
(b) cos^2(θ)
(c) cos(2θ)

Learn how to find the exact values of cos(-θ), cos^2(θ), and cos(2θ) for θ = 3π/4. This solution uses key trigonometric concepts, including angle properties and double-angle identities, to provide accurate calculations suitable for high school students in Grades 10-12.

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