If `cos(alpha + gamma)/cos(alpha - gamma) = cos 2beta` then `tan alpha, tan beta and tan gamma` are

Options

A. AP
B. GP
C. HP
D. none of these

Answer

To solve the equation

\[ \frac{\cos(\alpha + \gamma)}{\cos(\alpha - \gamma)} = \cos(2\beta) \]

we will use trigonometric identities step by step.

### Step 1: Apply the Cosine Addition and Subtraction Formulas
Using the formulas for cosine of sum and difference, we have:

\[ \cos(\alpha + \gamma) = \cos\alpha \cos\gamma - \sin\alpha \sin\gamma \]
\[ \cos(\alpha - \gamma) = \cos\alpha \cos\gamma + \sin\alpha \sin\gamma \]

### Step 2: Substitute into the Equation
Substituting these into the original equation gives:

\[ \frac{\cos\alpha \cos\gamma - \sin\alpha \sin\gamma}{\cos\alpha \cos\gamma + \sin\alpha \sin\gamma} = \cos(2\beta) \]

### Step 3: Use the Cosine Double Angle Formula
Recall that

\[ \cos(2\beta) = 2\cos^2\beta - 1 \]

Thus, we can rewrite the equation as:

\[ \frac{\cos\alpha \cos\gamma - \sin\alpha \sin\gamma}{\cos\alpha \cos\gamma + \sin\alpha \sin\gamma} = 2\cos^2\beta - 1 \]

### Step 4: Cross Multiply
Cross-multiplying gives:

\[ \cos\alpha \cos\gamma - \sin\alpha \sin\gamma = (2\cos^2\beta - 1)(\cos\alpha \cos\gamma + \sin\alpha \sin\gamma) \]

### Step 5: Expand the Right Side
Expanding the right side:

\[ \cos\alpha \cos\gamma - \sin\alpha \sin\gamma = (2\cos^2\beta \cos\alpha \cos\gamma + 2\cos^2\beta \sin\alpha \sin\gamma - \cos\alpha \cos\gamma - \sin\alpha \sin\gamma) \]

### Step 6: Rearranging Terms
Rearranging gives:

\[ \cos\alpha \cos\gamma - 2\cos^2\beta \cos\alpha \cos\gamma - 2\cos^2\beta \sin\alpha \sin\gamma + \sin\alpha \sin\gamma = 0 \]

### Step 7: Factor Out Common Terms
Factoring out common terms:

\[ \cos\alpha \cos\gamma(1 - 2\cos^2\beta) + \sin\alpha \sin\gamma(1 - 2\cos^2\beta) = 0 \]

### Step 8: Set Each Factor to Zero
This implies either:

1. \(1 - 2\cos^2\beta = 0\) or
2. \(\cos\alpha \cos\gamma + \sin\alpha \sin\gamma = 0\)

From the first equation, we find:

\[ \cos^2\beta = \frac{1}{2} \implies \tan^2\beta = 1 \implies \tan\beta = 1 \]

From the second equation, we have:

\[ \cos\alpha \cos\gamma + \sin\alpha \sin\gamma = 0 \implies \tan\alpha \tan\gamma = -1 \]

### Conclusion
Thus, we find that:

- \(\tan\alpha, \tan\beta, \tan\gamma\) are in **GP** (Geometric Progression).