We will first approximate $\cos x$ at $x = \frac{\pi}{4}$ using the given series expansion of cosine:

$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \cdots$

Step 1: Use the expansion up to $x^8$

We substitute $x = \frac{\pi}{4}$ (approximately 0.7853981633974483) into the series:

$\cos \left( \frac{\pi}{4} \right) = 1 - \frac{\left( \frac{\pi}{4} \right)^2}{2!} + \frac{\left( \frac{\pi}{4} \right)^4}{4!} - \frac{\left( \frac{\pi}{4} \right)^6}{6!} + \frac{\left( \frac{\pi}{4} \right)^8}{8!}$

Now, let's compute each term in the series.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can guide you through the calculations manually, or you can try another question. Here are some related topics you might want to explore:

1. How can I derive the Taylor series expansion for other trigonometric functions?
2. How does the error change when we include more terms in the series?
3. What is the difference between absolute and relative error?
4. How is the percentage error calculated in approximations?
5. Can you explain the factorial terms in the Taylor series?

Tip: The more terms you include in the Taylor series expansion, the closer your approximation gets to the actual value.