To derive the trigonometric identity \(\sin^2 x = \frac{1 - \cos(2x)}{2}\), we can use the double-angle formulas for sine and cosine. Here's a step-by-step derivation:

### Double-Angle Formulas
The double-angle formulas for sine and cosine are:
\[ \cos(2x) = \cos^2 x - \sin^2 x \]
\[ \sin(2x) = 2 \sin x \cos x \]

### Deriving the Identity for \(\sin^2 x\)
We start with the Pythagorean identity:
\[ \sin^2 x + \cos^2 x = 1 \]

From this, we can express \(\cos^2 x\) in terms of \(\sin^2 x\):
\[ \cos^2 x = 1 - \sin^2 x \]

Next, we use the double-angle formula for cosine:
\[ \cos(2x) = \cos^2 x - \sin^2 x \]

Substitute \(\cos^2 x\) from the Pythagorean identity:
\[ \cos(2x) = (1 - \sin^2 x) - \sin^2 x \]
\[ \cos(2x) = 1 - 2\sin^2 x \]

Rearranging this equation to solve for \(\sin^2 x\):
\[ 2\sin^2 x = 1 - \cos(2x) \]
\[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]

Thus, we have derived the identity:
\[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]

This identity is useful for integrating \(\sin^2 x\), as it allows us to convert the square of sine into a form involving \(\cos(2x)\), which is often easier to integrate.

Would you like more details on this derivation or any other topic? Here are 5 questions you might be interested in:

1. How do you derive the double-angle formulas?
2. Can you explain the Pythagorean identities in trigonometry?
3. How do you use trigonometric identities to simplify integrals?
4. What are other useful trigonometric identities for integration?
5. How can you use substitution to solve integrals involving trigonometric functions?

**Tip:** Familiarizing yourself with key trigonometric identities and their derivations can greatly simplify the process of solving integrals and other trigonometric problems.

Learn how to derive the trigonometric identity sin^2 x = (1 - cos(2x))/2 step-by-step. This problem covers key trigonometric concepts, including double-angle formulas and the Pythagorean identity. Suitable for advanced high school students and anyone studying trigonometry.

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