If `cosx = (1)/(2)( a+ (1)/(a))`, then cos3x is -

Options

A. `(1)/(2)(a^(3)+(1)/(a^(3)))`
B. `(3)/(2)(a^(3)+(1)/(a^(3)))`
C. `(1)/(2)(a^(3)-(1)/(a^(3)))`
D. `(3)/(2)(a^(3)-(1)/(a^(3)))`

Answer

To find the value of \( \cos 3x \) given that \( \cos x = \frac{1}{2} \left( a + \frac{1}{a} \right) \), we can use the trigonometric identity for \( \cos 3x \):

\[ \cos 3x = 4 \cos^3 x - 3 \cos x \]

### Step 1: Substitute the value of \( \cos x \)

Given \( \cos x = \frac{1}{2} \left( a + \frac{1}{a} \right) \), we can substitute this into the identity:

\[ \cos 3x = 4 \left( \frac{1}{2} \left( a + \frac{1}{a} \right) \right)^3 - 3 \left( \frac{1}{2} \left( a + \frac{1}{a} \right) \right) \]

### Step 2: Calculate \( \cos^3 x \)

First, we need to calculate \( \left( \frac{1}{2} \left( a + \frac{1}{a} \right) \right)^3 \):

\[ \left( \frac{1}{2} \left( a + \frac{1}{a} \right) \right)^3 = \frac{1}{8} \left( a + \frac{1}{a} \right)^3 \]

Using the binomial expansion:

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

So we can write:

\[ \left( a + \frac{1}{a} \right)^3 = a^3 + 3 \left( a + \frac{1}{a} \right) + \frac{1}{a^3} \]

### Step 3: Substitute back into the equation

Now substituting back into the equation for \( \cos 3x \):

\[ \cos 3x = 4 \cdot \frac{1}{8} \left( a^3 + 3 \left( a + \frac{1}{a} \right) + \frac{1}{a^3} \right) - 3 \cdot \frac{1}{2} \left( a + \frac{1}{a} \right) \]

This simplifies to:

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

### Step 4: Combine like terms

Now we combine the terms:

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

The \( 3 \left( a + \frac{1}{a} \right) \) terms cancel out:

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

### Final Result

Thus, the final expression for \( \cos 3x \) is:

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