Use power-reducing identities sin squared x equals fraction numerator 1 minus cos open parentheses 2 x close parentheses over denominator 2 end fraction to write an equivalent expression for sin to the power of 4 x with no exponent larger than 1.

To rewrite \(\sin^4{x}\) using power-reducing identities, we start by applying the identity for \(\sin^2{x}\):

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

Now, express \(\sin^4{x}\) as:

\[
\sin^4{x} = (\sin^2{x})^2
\]

Substitute the identity for \(\sin^2{x}\):

\[
\sin^4{x} = \left(\frac{1 - \cos(2x)}{2}\right)^2
\]

Expand the square:

\[
\sin^4{x} = \frac{(1 - \cos(2x))^2}{4}
\]

Now, expand \((1 - \cos(2x))^2\):

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

Thus:

\[
\sin^4{x} = \frac{1 - 2\cos(2x) + \cos^2(2x)}{4}
\]

Next, apply the power-reducing identity again to the \(\cos^2(2x)\) term:

\[
\cos^2(2x) = \frac{1 + \cos(4x)}{2}
\]

Substitute this into the expression:

\[
\sin^4{x} = \frac{1 - 2\cos(2x) + \frac{1 + \cos(4x)}{2}}{4}
\]

Simplify the expression:

\[
\sin^4{x} = \frac{1 - 2\cos(2x) + \frac{1}{2} + \frac{\cos(4x)}{2}}{4}
\]

Combine the constant terms:

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

Finally, distribute the \( \frac{1}{4} \):

\[
\sin^4{x} = \frac{3}{8} - \frac{\cos(2x)}{2} + \frac{\cos(4x)}{8}
\]

Thus, the equivalent expression for \(\sin^4{x}\) is:

\[
\sin^4{x} = \frac{3}{8} - \frac{\cos(2x)}{2} + \frac{\cos(4x)}{8}
\]

Would you like a detailed breakdown of any of the steps?

Here are 5 related questions to explore this further:
1. How can power-reducing identities be used to simplify \(\cos^4{x}\)?
2. What are other applications of power-reducing identities in trigonometry?
3. Can you derive the identity for \(\cos^2{x}\) using similar methods?
4. How do power-reducing identities relate to double-angle identities?
5. How would you integrate \(\sin^4{x}\) using this expanded form?

**Tip**: Power-reducing identities are particularly useful for reducing higher powers of trigonometric functions when solving integrals or simplifying expressions.

Learn how to rewrite sin^4(x) using power-reducing identities with a detailed step-by-step solution. This problem involves key trigonometric concepts such as double-angle identities and power-reducing identities. Suitable for high school students in Grades 10-12.

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