Which of the following is equivalent to: (9sin(x) + 9cos(x)) ^ 2 ?

To simplify the expression \((9\sin(x) + 9\cos(x))^2\), let's start by expanding the square using the distributive property:

\[
(9\sin(x) + 9\cos(x))^2 = (9\sin(x))^2 + 2(9\sin(x))(9\cos(x)) + (9\cos(x))^2
\]

Simplify each term:

1. \((9\sin(x))^2 = 81\sin^2(x)\)
2. \(2(9\sin(x))(9\cos(x)) = 2 \cdot 81\sin(x)\cos(x) = 162\sin(x)\cos(x)\)
3. \((9\cos(x))^2 = 81\cos^2(x)\)

Now, combine all terms:

\[
81\sin^2(x) + 162\sin(x)\cos(x) + 81\cos^2(x)
\]

Notice that \(81\sin^2(x) + 81\cos^2(x)\) can be simplified using the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\):

\[
81(\sin^2(x) + \cos^2(x)) = 81 \cdot 1 = 81
\]

Thus, the expression simplifies to:

\[
81 + 162\sin(x)\cos(x)
\]

Therefore, the equivalent expression to \((9\sin(x) + 9\cos(x))^2\) is:

\[
\boxed{81 + 162\sin(x)\cos(x)}
\]

Would you like further details on any part of this problem?

### Related Questions:
1. How do you expand binomials involving trigonometric functions?
2. What is the Pythagorean identity in trigonometry?
3. How can we express \(\sin(2x)\) in terms of \(\sin(x)\) and \(\cos(x)\)?
4. What is the double-angle formula for sine?
5. How do you simplify expressions involving \(\sin(x)\) and \(\cos(x)\)?

**Tip:** When dealing with trigonometric expressions, always look for opportunities to apply fundamental identities like the Pythagorean identity.

Which of the following is equivalent to: (9sin(x) + 9cos(x)) ^ 2?

This guide walks through the simplification of the expression (9sin(x) + 9cos(x))^2, demonstrating the application of trigonometric identities such as the Pythagorean identity and the double-angle formula for sine. Suitable for high school students, Grades 10-12.

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