The expression given is:

$\cos^2\left(\frac{\pi}{12}\right) + \sin^2\left(\frac{\pi}{12}\right)$

This is a direct application of the Pythagorean identity:

$\cos^2 \theta + \sin^2 \theta = 1$

for any angle $\theta$.

Therefore:

$\cos^2\left(\frac{\pi}{12}\right) + \sin^2\left(\frac{\pi}{12}\right) = 1$

So the value of this expression is $\boxed{1}$.

Would you like a further explanation of the Pythagorean identity or help with another problem?

Related Questions:

1. How do you derive the Pythagorean identity from the unit circle?
2. What is the exact value of $\cos\left(\frac{\pi}{12}\right)$ and $\sin\left(\frac{\pi}{12}\right)$?
3. How do we use the double angle formulas to compute trigonometric values?
4. How would you prove other trigonometric identities, like the sum and difference formulas?
5. What are some real-world applications of the Pythagorean identity?

Tip: The Pythagorean identity is fundamental in trigonometry and can be used to simplify many expressions, especially when working with sine and cosine functions.