The half-angle formula for $\sin^4 \theta$ can be derived using the half-angle identity for sine. Let's break it down step by step:

Step 1: Start with the basic half-angle identity for sine:

$\sin^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{2}$

Step 2: Express $\sin^4\left(\frac{\theta}{2}\right)$:

Since we are looking for $\sin^4\left(\frac{\theta}{2}\right)$, we square the half-angle identity: $\sin^4\left(\frac{\theta}{2}\right) = \left(\frac{1 - \cos\theta}{2}\right)^2$

Step 3: Simplify the expression:

$\sin^4\left(\frac{\theta}{2}\right) = \frac{(1 - \cos\theta)^2}{4}$

Thus, the formula for $\sin^4\left(\frac{\theta}{2}\right)$ is: $\sin^4\left(\frac{\theta}{2}\right) = \frac{1 - 2\cos\theta + \cos^2\theta}{4}$

This is the half-angle expression for $\sin^4\left(\frac{\theta}{2}\right)$.

Would you like further details or clarifications?

Here are 5 questions to expand on the concept:

1. What is the half-angle formula for $\cos^2\theta$?
2. How do we derive the double-angle identity for sine?
3. How can the expression for $\sin^4 \theta$ be integrated over an interval?
4. What are some real-world applications of the half-angle identities?
5. How does $\sin^4\theta$ behave as $\theta$ approaches 0 or $\pi$?

Tip: Half-angle formulas are especially useful in simplifying trigonometric integrals and solving equations where direct application of standard identities may be complex.