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.

Explore the expansion of the expression [(e^it - e^-it)/2]^4 using Euler's formula and trigonometric identities. This detailed guide explains the step-by-step process, including power reduction formulas for sin^4(t), suitable for undergraduate mathematics students.

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