1. What is equivalent to the expression, sin^3x?

2. What is equivalent to the expression, sin^3x cos^3x?


Using Power Reduced Identities

To express \(\sin^3(x)\) and \(\sin^3(x) \cos^3(x)\) using power-reducing identities, let's go through each step.

### 1. Simplifying \(\sin^3(x)\)

The power-reducing identity for \(\sin^2(x)\) is:

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

Thus, we can write \(\sin^3(x)\) as:

\[
\sin^3(x) = \sin(x) \cdot \sin^2(x)
\]

Substituting the identity for \(\sin^2(x)\):

\[
\sin^3(x) = \sin(x) \cdot \frac{1 - \cos(2x)}{2}
\]

Expanding this gives:

\[
\sin^3(x) = \frac{\sin(x) - \sin(x) \cos(2x)}{2}
\]

Now, using the identity \(\sin(x) \cos(2x) = \frac{\sin(3x) + \sin(x)}{2}\), we can further simplify:

\[
\sin^3(x) = \frac{\sin(x)}{2} - \frac{\sin(3x) + \sin(x)}{4}
\]

Combining terms, we get:

\[
\sin^3(x) = \frac{3\sin(x)}{4} - \frac{\sin(3x)}{4}
\]

### 2. Simplifying \(\sin^3(x) \cos^3(x)\)

We can express \(\sin^3(x) \cos^3(x)\) as \((\sin(x) \cos(x))^3\).

Using the double-angle identity \(\sin(x) \cos(x) = \frac{\sin(2x)}{2}\), we get:

\[
\sin^3(x) \cos^3(x) = \left(\frac{\sin(2x)}{2}\right)^3
\]

Simplifying further:

\[
\sin^3(x) \cos^3(x) = \frac{\sin^3(2x)}{8}
\]

Now, to reduce \(\sin^3(2x)\) using the same approach as in part 1, we have:

\[
\sin^3(2x) = \frac{3\sin(2x)}{4} - \frac{\sin(6x)}{4}
\]

Substituting back:

\[
\sin^3(x) \cos^3(x) = \frac{3\sin(2x)}{32} - \frac{\sin(6x)}{32}
\]

### Final Answers

1. \(\sin^3(x) = \frac{3\sin(x)}{4} - \frac{\sin(3x)}{4}\)

2. \(\sin^3(x) \cos^3(x) = \frac{3\sin(2x)}{32} - \frac{\sin(6x)}{32}\)

Would you like further details on these simplifications or any other questions?

---

1. How would \(\cos^3(x)\) be expressed using power-reducing identities?
2. What if we had \(\sin^4(x)\) instead of \(\sin^3(x)\); how would we simplify that?
3. Can \(\sin^5(x)\) be simplified using power-reducing identities?
4. How would you express \(\cos^3(x) \sin^3(x)\) differently, perhaps using sum or product identities?
5. Could we use complex exponentials to represent \(\sin^3(x)\) or \(\sin^3(x) \cos^3(x)\) more simply?

**Tip:** Power-reducing identities are particularly useful when integrating or simplifying trigonometric powers in calculus, as they convert higher powers into more manageable forms.

What is equivalent to the expression, sin^3x? What is equivalent to the expression, sin^3x cos^3x?

Learn how to simplify the expressions sin^3(x) and sin^3(x)cos^3(x) using power-reducing identities. The solution includes step-by-step reductions and trigonometric transformations. This topic is ideal for high school students studying trigonometry.

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