sen elevado al 2 = ( 1 - cos x)(1 + cos x)

The equation you've given is:

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

Let's simplify the right-hand side to see if it equals \(\sin^2(x)\).

1. **Expand the right-hand side:**

   Use the difference of squares formula:

   \[
   (1 - \cos(x))(1 + \cos(x)) = 1^2 - \cos^2(x) = 1 - \cos^2(x)
   \]

2. **Recall the Pythagorean identity:**

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

   From this identity, we can express \(\sin^2(x)\) as:

   \[
   \sin^2(x) = 1 - \cos^2(x)
   \]

3. **Substitute into the equation:**

   \[
   \sin^2(x) = 1 - \cos^2(x)
   \]

This shows that the equation you provided is indeed correct:

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

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Do you want more details, or do you have any questions about this solution?

Here are 5 related questions you might find interesting:

1. How can \(\sin^2(x)\) be expressed in terms of \(\tan(x)\)?
2. What is the expansion of \((\sin(x) + \cos(x))^2\)?
3. How does the identity \(\cos^2(x) - \sin^2(x) = \cos(2x)\) relate to this equation?
4. Can you prove the identity \(\sin^4(x) + \cos^4(x) = 1 - 2\sin^2(x)\cos^2(x)\)?
5. What is the relationship between \(\sin^2(x)\) and \(\cos^2(x)\) when \(x = \frac{\pi}{4}\)?

**Tip:** Trigonometric identities like \(\sin^2(x) + \cos^2(x) = 1\) are very powerful and often serve as the basis for proving other identities.

Explore the equation sin^2(x) = (1 - cos(x))(1 + cos(x)) and its solution step-by-step. This problem delves into advanced trigonometric identities, including the difference of squares and the Pythagorean identity. Suitable for advanced high school students and those studying trigonometry.

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