If `cos theta = 1/2 (a+(1)/(a)),` then the value of `cos 3 theta ` is

Options

A. `1/8 (a ^(3) +(1)/(a ^(3)))`
B. `3/2 (a+(1)/(a))`
C. `1/2 (a^(3) + (1)/(a^(3)))`
D. `1/3 (a^(3) +(1)/(a ^(3)))`

Answer

To find the value of \( \cos 3\theta \) given that \( \cos \theta = \frac{1}{2} \left( a + \frac{1}{a} \right) \), we will follow these steps:

### Step 1: Use the formula for \( \cos 3\theta \)
The formula for \( \cos 3\theta \) is:
\[ \cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta \]

### Step 2: Substitute \( \cos \theta \)
Substituting the given value of \( \cos \theta \):
\[ \cos 3\theta = 4 \left( \frac{1}{2} \left( a + \frac{1}{a} \right) \right)^3 - 3 \left( \frac{1}{2} \left( a + \frac{1}{a} \right) \right) \]

### Step 3: Simplify the first term
Calculating \( \left( \frac{1}{2} \left( a + \frac{1}{a} \right) \right)^3 \):
\[ \left( \frac{1}{2} \left( a + \frac{1}{a} \right) \right)^3 = \frac{1}{8} \left( a + \frac{1}{a} \right)^3 \]
Thus,
\[ \cos 3\theta = 4 \cdot \frac{1}{8} \left( a + \frac{1}{a} \right)^3 - 3 \cdot \frac{1}{2} \left( a + \frac{1}{a} \right) \]
This simplifies to:
\[ \cos 3\theta = \frac{1}{2} \left( a + \frac{1}{a} \right)^3 - \frac{3}{2} \left( a + \frac{1}{a} \right) \]

### Step 4: Factor out \( \frac{1}{2} \)
Factoring out \( \frac{1}{2} \):
\[ \cos 3\theta = \frac{1}{2} \left( \left( a + \frac{1}{a} \right)^3 - 3 \left( a + \frac{1}{a} \right) \right) \]

### Step 5: Use the identity for \( (x+y)^3 \)
Using the identity \( x^3 + y^3 + 3xy(x+y) \):
Let \( x = a \) and \( y = \frac{1}{a} \):
\[ \left( a + \frac{1}{a} \right)^3 = a^3 + \frac{1}{a^3} + 3 \cdot a \cdot \frac{1}{a} \left( a + \frac{1}{a} \right) \]
This simplifies to:
\[ \left( a + \frac{1}{a} \right)^3 = a^3 + \frac{1}{a^3} + 3 \left( a + \frac{1}{a} \right) \]

### Step 6: Substitute back into the equation
Substituting this back into our equation:
\[ \cos 3\theta = \frac{1}{2} \left( a^3 + \frac{1}{a^3} + 3 \left( a + \frac{1}{a} \right) - 3 \left( a + \frac{1}{a} \right) \right) \]
The \( 3 \left( a + \frac{1}{a} \right) \) terms cancel out:
\[ \cos 3\theta = \frac{1}{2} \left( a^3 + \frac{1}{a^3} \right) \]

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

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