The expression \( 1 - \cos(2a) \) is commonly encountered in trigonometric identities. One of the standard trigonometric identities for \( \cos(2a) \) is the **double angle identity**:

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

Thus, you can rewrite \( 1 - \cos(2a) \) as:

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

So, the simplified form of \( 1 - \cos(2a) \) is:

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

This identity is useful in many trigonometric and calculus problems.

Would you like more details on how to use this identity in specific problems?

Here are some related questions you might find interesting:
1. What is the derivation of the double angle identities for sine and cosine?
2. How can \( 1 - \cos(2a) \) be applied in solving integrals?
3. What are other forms of \( \cos(2a) \) in terms of \( \sin(a) \) and \( \tan(a) \)?
4. How can we use this identity in proving trigonometric equations?
5. What is the relationship between \( \cos(2a) \) and the Pythagorean identity?

**Tip:** When solving trigonometric identities, it is often helpful to remember alternative forms of key functions like \( \cos(2a) \) in terms of \( \sin(a) \) or \( \tan(a) \).

Explore how to simplify the expression 1 - cos(2a) using the double angle identity for cosine. This solution breaks down key trigonometric concepts and provides a clear step-by-step process. Ideal for high school students in Grades 10-12 learning trigonometric identities.

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