To solve \( \cos^2\left(-\frac{\pi}{4}\right) \), let's break it down step by step.

1. **Cosine function at \( -\frac{\pi}{4} \):**

   The cosine function is an **even function**, meaning \( \cos(-x) = \cos(x) \). So, we can rewrite:
   \[
   \cos\left(-\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right)
   \]

2. **Cosine of \( \frac{\pi}{4} \):**

   From the unit circle, we know that:
   \[
   \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
   \]

3. **Square the result:**

   Now, square the cosine value:
   \[
   \cos^2\left(\frac{\pi}{4}\right) = \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} = \frac{1}{2}
   \]

Thus, the final answer is:
\[
\cos^2\left(-\frac{\pi}{4}\right) = \frac{1}{2}
\]

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. What is \( \sin^2\left(\frac{\pi}{4}\right) \)?
2. How do even and odd functions behave in trigonometry?
3. What is the value of \( \cos^2(0) \)?
4. How does the Pythagorean identity relate to cosine and sine?
5. Can you find \( \cos^2\left(\frac{3\pi}{4}\right) \)?

**Tip:** Remember that the cosine function is even, so \( \cos(-x) = \cos(x) \), which simplifies many trigonometric problems!

Learn how to solve cos^2(-pi/4) with a step-by-step explanation using the properties of even functions in trigonometry. Understand the cosine value for pi/4 and how to square it. Suitable for students in Grades 9-12, covering key trigonometric concepts and formulas.

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