If `sqrtx + (1)/( sqrtx) = 2 cos theta, ` then `x ^(6) +x^(-6)=`

Options

A. `2 cos 6 theta`
B. `2 cos 12theta`
C. `2 cos 3 theta`
D. `2 sin 3 theta`

Answer

To solve the problem, we start with the equation given:

\[ \sqrt{x} + \frac{1}{\sqrt{x}} = 2 \cos \theta \]

### Step 1: Square both sides
We will square both sides of the equation to eliminate the square root.

\[ \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2 = (2 \cos \theta)^2 \]

This expands to:

\[ x + 2 + \frac{1}{x} = 4 \cos^2 \theta \]

### Step 2: Rearrange the equation
Now, we can rearrange the equation to isolate \(x + \frac{1}{x}\):

\[ x + \frac{1}{x} = 4 \cos^2 \theta - 2 \]

### Step 3: Factor out the 2
We can factor out a 2 from the right side:

\[ x + \frac{1}{x} = 2(2 \cos^2 \theta - 1) \]

### Step 4: Use the double angle formula
Using the double angle formula for cosine, \(2 \cos^2 \theta - 1 = \cos(2\theta)\):

\[ x + \frac{1}{x} = 2 \cos(2\theta) \]

### Step 5: Square again
Next, we square both sides again to find \(x^2 + \frac{1}{x^2}\):

\[ \left(x + \frac{1}{x}\right)^2 = (2 \cos(2\theta))^2 \]

This expands to:

\[ x^2 + 2 + \frac{1}{x^2} = 4 \cos^2(2\theta) \]

### Step 6: Rearrange for \(x^2 + \frac{1}{x^2}\)
Rearranging gives us:

\[ x^2 + \frac{1}{x^2} = 4 \cos^2(2\theta) - 2 \]

### Step 7: Factor out the 2 again
Factoring out a 2:

\[ x^2 + \frac{1}{x^2} = 2(2 \cos^2(2\theta) - 1) \]

### Step 8: Use the double angle formula again
Using the double angle formula again, we have:

\[ x^2 + \frac{1}{x^2} = 2 \cos(4\theta) \]

### Step 9: Cube to find \(x^6 + \frac{1}{x^6}\)
Now, we will use the identity for cubes:

\[ \left(x^2 + \frac{1}{x^2}\right)^3 = x^6 + \frac{1}{x^6} + 3\left(x^2 + \frac{1}{x^2}\right) \]

Substituting \(x^2 + \frac{1}{x^2}\):

\[ (2 \cos(4\theta))^3 = x^6 + \frac{1}{x^6} + 3(2 \cos(4\theta)) \]

### Step 10: Calculate \(x^6 + \frac{1}{x^6}\)
Calculating \( (2 \cos(4\theta))^3 \):

\[ 8 \cos^3(4\theta) = x^6 + \frac{1}{x^6} + 6 \cos(4\theta) \]

Rearranging gives us:

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

### Step 11: Factor out the common term
Factoring out \(2 \cos(4\theta)\):

\[ x^6 + \frac{1}{x^6} = 2 \cos(4\theta)(4 \cos^2(4\theta) - 3) \]

### Step 12: Use the triple angle formula
Using the triple angle formula for cosine, we have:

\[ x^6 + \frac{1}{x^6} = 2 \cos(12\theta) \]

### Final Answer
Thus, the final answer is:

\[ x^6 + \frac{1}{x^6} = 2 \cos(12\theta) \]

---