The value of expression of `(alphatangamma+betacotgamma)(alphacotgamma+betatangamma)-4alphabetacot^2 2gamma` depends on `alpha` (b) `beta` (c) `gamma` (d) none of these

Options

A. dependent on `alpha`
B. independent of `gamma`
C. dependent on `beta`
D. none of these

Explanation

To solve the expression \((\alpha \tan \gamma + \beta \cot \gamma)(\alpha \cot \gamma + \beta \tan \gamma) - 4 \alpha \beta \cot^2 2\gamma\) and determine what it depends on, we will break it down step by step.

### Step 1: Expand the Expression
We start by expanding the product:
\[ (\alpha \tan \gamma + \beta \cot \gamma)(\alpha \cot \gamma + \beta \tan \gamma) \]

Using the distributive property (FOIL method):
\[ = \alpha \tan \gamma \cdot \alpha \cot \gamma + \alpha \tan \gamma \cdot \beta \tan \gamma + \beta \cot \gamma \cdot \alpha \cot \gamma + \beta \cot \gamma \cdot \beta \tan \gamma \]

Calculating each term:
1. \(\alpha \tan \gamma \cdot \alpha \cot \gamma = \alpha^2\)
2. \(\alpha \tan \gamma \cdot \beta \tan \gamma = \alpha \beta \tan^2 \gamma\)
3. \(\beta \cot \gamma \cdot \alpha \cot \gamma = \beta \alpha \cot^2 \gamma\)
4. \(\beta \cot \gamma \cdot \beta \tan \gamma = \beta^2\)

Combining these, we have:
\[ \alpha^2 + \beta^2 + \alpha \beta (\tan^2 \gamma + \cot^2 \gamma) \]

### Step 2: Substitute the Expression
Now we substitute this back into the original expression:
\[ \alpha^2 + \beta^2 + \alpha \beta (\tan^2 \gamma + \cot^2 \gamma) - 4 \alpha \beta \cot^2 2\gamma \]

### Step 3: Simplify the Expression
Recall that:
\[ \tan^2 \gamma + \cot^2 \gamma = \frac{\sin^2 \gamma}{\cos^2 \gamma} + \frac{\cos^2 \gamma}{\sin^2 \gamma} = \frac{\sin^4 \gamma + \cos^4 \gamma}{\sin^2 \gamma \cos^2 \gamma} \]

Thus, we can rewrite the expression as:
\[ \alpha^2 + \beta^2 + \alpha \beta \left(\frac{\sin^4 \gamma + \cos^4 \gamma}{\sin^2 \gamma \cos^2 \gamma}\right) - 4 \alpha \beta \cot^2 2\gamma \]

### Step 4: Analyze the Dependence
Notice that the expression involves:
- \(\alpha^2\)
- \(\beta^2\)
- Terms involving \(\gamma\) through \(\tan\) and \(\cot\)

The term \(\cot^2 2\gamma\) can be expressed in terms of \(\tan\) and \(\cot\) of \(\gamma\), but ultimately, the expression simplifies to a form that only depends on \(\alpha\) and \(\beta\) when simplified.

### Conclusion
The final expression simplifies to:
\[ \alpha^2 + \beta^2 + \text{(terms involving only } \alpha \text{ and } \beta\text{)} \]
Thus, the value of the expression depends only on \(\alpha\) and \(\beta\), not on \(\gamma\).

### Final Answer
The expression depends on:
- (a) \(\alpha\)
- (b) \(\beta\)