find the exact value of cos^(2)65(degrees)+sin^(2)65(degrees)

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.

Explore how to find the exact value of cos^2(65°) + sin^2(65°) using the Pythagorean identity in trigonometry. This solution explains the use of the identity cos^2(θ) + sin^2(θ) = 1, which holds true for all angles. Perfect for students in Grades 9-10 studying trigonometric identities.

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