If `x+(1)/(x)=2costheta`, then find the valueof `x^(3)+(1)/(x^(3))`.

Options

A. `2cos3theta`
B. `2sin3theta`
C. `sin3theta`
D. `cos3theta`

Answer

To solve the problem, we need to find the value of \( x^3 + \frac{1}{x^3} \) given that \( x + \frac{1}{x} = 2 \cos \theta \).

### Step-by-step Solution:

1. **Start with the given equation:**
\[ x + \frac{1}{x} = 2 \cos \theta \]

2. **Cube both sides:**
\[ \left( x + \frac{1}{x} \right)^3 = (2 \cos \theta)^3 \]
This simplifies to:
\[ x^3 + 3x\left(\frac{1}{x}\right)\left(x + \frac{1}{x}\right) + \frac{1}{x^3} = 8 \cos^3 \theta \]
Since \( x\left(\frac{1}{x}\right) = 1 \), we can simplify further:
\[ x^3 + \frac{1}{x^3} + 3\left(x + \frac{1}{x}\right) = 8 \cos^3 \theta \]

3. **Substitute \( x + \frac{1}{x} \) with \( 2 \cos \theta \):**
\[ x^3 + \frac{1}{x^3} + 3(2 \cos \theta) = 8 \cos^3 \theta \]
This simplifies to:
\[ x^3 + \frac{1}{x^3} + 6 \cos \theta = 8 \cos^3 \theta \]

4. **Isolate \( x^3 + \frac{1}{x^3} \):**
\[ x^3 + \frac{1}{x^3} = 8 \cos^3 \theta - 6 \cos \theta \]

5. **Factor out \( 2 \cos \theta \):**
\[ x^3 + \frac{1}{x^3} = 2 \cos \theta (4 \cos^2 \theta - 3) \]

6. **Recognize the identity:**
The expression \( 4 \cos^2 \theta - 3 \) can be recognized as \( \cos 3\theta \):
\[ x^3 + \frac{1}{x^3} = 2 \cos \theta \cos 3\theta \]

### Final Result:
Thus, the value of \( x^3 + \frac{1}{x^3} \) is:
\[ \boxed{2 \cos 3\theta} \]