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The expression you provided is:

\[
5 \cos^2 a + 5 \sin^2 a - 3
\]

We can simplify this by using the Pythagorean identity, which states:

\[
\cos^2 a + \sin^2 a = 1
\]

Let's apply this identity to the given expression:

\[
5(\cos^2 a + \sin^2 a) - 3 = 5(1) - 3 = 5 - 3 = 2
\]

So, the simplified expression is:

\[
2
\]

Let me know if you need further details!

Here are five related questions for further exploration:
1. How do we derive the Pythagorean identity \(\cos^2 a + \sin^2 a = 1\)?
2. Can the expression \( \cos^2 a + \sin^2 a = 1 \) be used in other trigonometric simplifications?
3. What is the geometric interpretation of the Pythagorean identity in the unit circle?
4. How would the original expression change if we had different coefficients for \(\cos^2 a\) and \(\sin^2 a\)?
5. Can you generalize this process for simplifying similar expressions involving trigonometric identities?

**Tip:** Always look out for key identities like the Pythagorean identity when simplifying trigonometric expressions—they often help simplify complex terms!

Learn how to simplify the trigonometric expression 5cos^2(a) + 5sin^2(a) - 3 using the Pythagorean identity cos^2(a) + sin^2(a) = 1. This problem covers fundamental trigonometric concepts, making it suitable for students in Grades 9-11.

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