If x+\frac{1}{x}=2~\cos~\theta, then what is x^{3}+\frac{1}{x^{3}} equal to?
- A. \cos^{3}\theta
- B. \cos~3\theta
- C. 2~\cos~3\theta ✓
- D. 3~\cos~3\theta
Correct Answer: C. 2~\cos~3\theta
Explanation
Use the algebraic identity a^3 + b^3 = (a+b)^3 - 3ab(a+b). Substitute x and \frac{1}{x} to get x^3 + \frac{1}{x^3} = (2\cos\theta)^3 - 3(1)(2\cos\theta). This yields 8\cos^3\theta - 6\cos\theta = 2(4\cos^3\theta - 3\cos\theta). Using the multiple angle formula \cos 3\theta = 4\cos^3\theta - 3\cos\theta, the result simplifies to 2\cos 3\theta.
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