Question

If $\cos \theta = \frac{1}{2}(a + \frac{1}{a})$, then $4 \cos^3 \theta - 3 \cos \theta =$

• A. $a^3 + \frac{1}{a^3}$
• B. $\frac{1}{2}(a^3 + \frac{1}{a^3})$
• C. $\frac{1}{4}(a^3 + \frac{1}{a^3})$
• D. $\frac{1}{3}(a^3 + \frac{1}{a^3})$

Hint:

The identity $4\cos^3\theta - 3\cos\theta$ is simply $\cos 3\theta$.

Answer 1

The Correct Option is B

Step 1: Concept
Recognize the triple angle formula: $\cos 3\theta = 4\cos^3 \theta - 3\cos \theta$.

Step 2: Meaning
We need to calculate $\cos 3\theta$ given the value of $\cos \theta$.

Step 3: Analysis
If $\cos \theta = \frac{1}{2}(e^{i\theta} + e^{-i\theta})$, then let $a = e^{i\theta}$. Then $\cos 3\theta = \frac{1}{2}(e^{i3\theta} + e^{-i3\theta})$.

Step 4: Conclusion
Substituting $a^3$ for $e^{i3\theta}$, we get $\cos 3\theta = \frac{1}{2}(a^3 + 1/a^3)$.
Final Answer: (B)