Step 1: Understanding the Question:
The question asks to find the value of a specific cubic expression in \( \cos \theta \) given an expression for \( \cos \theta \) in terms of a variable \( a \).
Step 2: Key Formula or Approach:
1. Identify the cubic expression as the triple angle formula: \( \cos 3\theta = 4\cos^3 \theta - 3\cos \theta \).
2. Use De Moivre's Theorem concepts where \( \cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2} \).
Step 3: Detailed Explanation:
Let's associate the variable \( a \) with the complex exponential \( e^{i\theta} \).
If \( a = e^{i\theta} \), then \( \frac{1}{a} = e^{-i\theta} \).
Then \( \frac{1}{2}(a + 1/a) = \frac{e^{i\theta} + e^{-i\theta}}{2} \), which is exactly the definition of \( \cos \theta \).
We need to find \( \cos 3\theta \).
Using the same logic, if \( a = e^{i\theta} \), then \( a^3 = (e^{i\theta})^3 = e^{i3\theta} \) and \( \frac{1}{a^3} = e^{-i3\theta} \).
By definition, \( \cos 3\theta = \frac{e^{i3\theta} + e^{-i3\theta}}{2} \).
Substituting the terms of \( a \):
\[ \cos 3\theta = \frac{1}{2}\left(a^3 + \frac{1}{a^3}\right) \]
Since \( \cos 3\theta = 4\cos^3 \theta - 3\cos \theta \), the answer is:
\[ \frac{1}{2}\left(a^3 + \frac{1}{a^3}\right) \]
Step 4: Final Answer:
The result of the expression is \( \frac{1}{2}\left(a^3 + \frac{1}{a^3}\right) \).