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The image contains a mathematical problem asking to prove that \(\sin(3\alpha) = 3\sin\alpha - 4\sin^3\alpha\).

### Proof:

We start with the left-hand side (LHS) of the equation:

\[
\sin(3\alpha)
\]

We can use the trigonometric identity for the sine of a triple angle:

\[
\sin(3\alpha) = 3\sin\alpha - 4\sin^3\alpha
\]

**Derivation of the identity:**

To derive the identity \(\sin(3\alpha) = 3\sin\alpha - 4\sin^3\alpha\), we use the angle sum identity:

\[
\sin(3\alpha) = \sin(2\alpha + \alpha)
\]

Using the sine angle addition formula:

\[
\sin(2\alpha + \alpha) = \sin 2\alpha \cos \alpha + \cos 2\alpha \sin \alpha
\]

Now, recall the double-angle identities:

\[
\sin 2\alpha = 2\sin \alpha \cos \alpha
\]
\[
\cos 2\alpha = 1 - 2\sin^2 \alpha
\]

Substitute these into the expression:

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

Simplify:

\[
\sin(3\alpha) = 2\sin \alpha \cos^2 \alpha + \sin \alpha - 2\sin^3 \alpha
\]

Use the identity \(\cos^2 \alpha = 1 - \sin^2 \alpha\):

\[
\sin(3\alpha) = 2\sin \alpha (1 - \sin^2 \alpha) + \sin \alpha - 2\sin^3 \alpha
\]
\[
\sin(3\alpha) = 2\sin \alpha - 2\sin^3 \alpha + \sin \alpha - 2\sin^3 \alpha
\]
\[
\sin(3\alpha) = 3\sin \alpha - 4\sin^3 \alpha
\]

Thus, the identity is proven.

### Do you want more details or have any questions? Here are some related questions you might find useful:

1. How does the identity for \(\cos(3\alpha)\) relate to the one for \(\sin(3\alpha)\)?
2. Can you prove the identity \(\cos(3\alpha) = 4\cos^3\alpha - 3\cos\alpha\)?
3. What is the geometric interpretation of the triple angle identity?
4. How would you use the identity \(\sin(3\alpha)\) in solving trigonometric equations?
5. How does this identity simplify the calculation of trigonometric values for specific angles?

**Tip:** Understanding and proving trigonometric identities is crucial in simplifying complex trigonometric expressions and solving equations efficiently.

Prove that \(\sin(3\alpha) = 3\sin\alpha - 4\sin^3\alpha\).

Learn how to prove the trigonometric identity \(\sin(3\alpha) = 3\sin\alpha - 4\sin^3\alpha\) with a step-by-step solution. This problem covers key trigonometric concepts and identities, suitable for students in Grades 10-12.

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