The expression $\cos^2(65^\circ) + \sin^2(65^\circ)$ can be simplified using the Pythagorean identity:

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

This identity holds true for any angle $\theta$. In this case, since $\theta = 65^\circ$, we can conclude:

$\cos^2(65^\circ) + \sin^2(65^\circ) = 1$

So, the exact value is:

$1$

Would you like further clarification or details on this?

Here are 5 related questions to explore further:

1. What is the Pythagorean identity in trigonometry?
2. How can you prove $\cos^2(\theta) + \sin^2(\theta) = 1$ geometrically?
3. How does the Pythagorean identity apply to non-acute angles?
4. What are the other trigonometric identities, such as sum and difference formulas?
5. How do you calculate $\cos(65^\circ)$ or $\sin(65^\circ)$ using a calculator?

Tip: Always remember that trigonometric identities are fundamental tools for simplifying and solving many types of trigonometric equations.