If `cos alpha + cos beta + cos gamma = 0 `then `cos 3alpha + cos 3beta + cos 3gamma = lambda cos alpha *cos beta*cos gamma` where`lambda` is equal to

Options

A. 12
B. 4
C. 1
D. none of these

Answer

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

\[ \cos \alpha + \cos \beta + \cos \gamma = 0 \]

We need to find the value of \(\lambda\) such that:

\[ \cos 3\alpha + \cos 3\beta + \cos 3\gamma = \lambda \cos \alpha \cos \beta \cos \gamma \]

### Step 1: Use the formula for \(\cos 3\theta\)

We know the formula for \(\cos 3\theta\):

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

Using this formula, we can express \(\cos 3\alpha\), \(\cos 3\beta\), and \(\cos 3\gamma\):

\[ \cos 3\alpha = 4\cos^3 \alpha - 3\cos \alpha \]
\[ \cos 3\beta = 4\cos^3 \beta - 3\cos \beta \]
\[ \cos 3\gamma = 4\cos^3 \gamma - 3\cos \gamma \]

### Step 2: Combine the expressions

Now, we can combine these expressions:

\[ \cos 3\alpha + \cos 3\beta + \cos 3\gamma = (4\cos^3 \alpha - 3\cos \alpha) + (4\cos^3 \beta - 3\cos \beta) + (4\cos^3 \gamma - 3\cos \gamma) \]

This simplifies to:

\[ = 4(\cos^3 \alpha + \cos^3 \beta + \cos^3 \gamma) - 3(\cos \alpha + \cos \beta + \cos \gamma) \]

### Step 3: Substitute the known value

Since we know that \(\cos \alpha + \cos \beta + \cos \gamma = 0\), we can substitute this into our equation:

\[ = 4(\cos^3 \alpha + \cos^3 \beta + \cos^3 \gamma) - 3 \cdot 0 \]

Thus, we have:

\[ \cos 3\alpha + \cos 3\beta + \cos 3\gamma = 4(\cos^3 \alpha + \cos^3 \beta + \cos^3 \gamma) \]

### Step 4: Use the identity for the sum of cubes

We can use the identity for the sum of cubes:

\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

Let \(a = \cos \alpha\), \(b = \cos \beta\), \(c = \cos \gamma\). Thus, we have:

\[ \cos^3 \alpha + \cos^3 \beta + \cos^3 \gamma = 3\cos \alpha \cos \beta \cos \gamma \]

### Step 5: Substitute back into the equation

Now substituting this back into our equation:

\[ \cos 3\alpha + \cos 3\beta + \cos 3\gamma = 4(3 \cos \alpha \cos \beta \cos \gamma) \]

This simplifies to:

\[ \cos 3\alpha + \cos 3\beta + \cos 3\gamma = 12 \cos \alpha \cos \beta \cos \gamma \]

### Step 6: Identify \(\lambda\)

From the equation, we can see that:

\[ \lambda = 12 \]

Thus, the value of \(\lambda\) is:

\[ \boxed{12} \]