To expand the expression $\left( \frac{e^{it} - e^{-it}}{2} \right)^4$, let's proceed step-by-step.

1. Recognize that $\frac{e^{it} - e^{-it}}{2}$ is equivalent to $\sin(t)$, based on Euler's formula: $e^{it} = \cos(t) + i\sin(t) \quad \text{and} \quad e^{-it} = \cos(t) - i\sin(t)$ Therefore: $\frac{e^{it} - e^{-it}}{2} = \sin(t)$

2. Now, substitute $\sin(t)$ into the expression: $\left( \frac{e^{it} - e^{-it}}{2} \right)^4 = \sin^4(t)$

3. Next, use the power reduction identity for $\sin^4(t)$: $\sin^4(t) = \frac{3}{8} - \frac{1}{2} \cos(2t) + \frac{1}{8} \cos(4t)$

Thus, the expanded form of $\left( \frac{e^{it} - e^{-it}}{2} \right)^4$ is: $\sin^4(t) = \frac{3}{8} - \frac{1}{2} \cos(2t) + \frac{1}{8} \cos(4t)$

Would you like more details on how the power reduction formula is derived or any other part of the process?

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Here are five related questions to explore further:

1. How do you derive the power reduction formula for $\sin^n(t)$?
2. What is Euler's formula and how does it apply to complex exponentials?
3. How can you expand $\sin^n(t)$ for general powers $n$?
4. Can you derive similar expansions for $\cos^4(t)$?
5. What is the relationship between $\sin^2(t)$ and $\cos(2t)$?

Tip: When expanding trigonometric powers, using power-reduction formulas is often faster than expanding everything manually.