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.