Let `sin^(-1)alpha+sin^(-1)beta+sin^(-1)gamma=pi` and `alpha, beta, gamma` are non zero real numbers such that `(alpha+beta+gamma)(alpha+beta-gamma)=3alphabeta` then value of `gamma`

Options

A. 1
B. `1/2`
C. `-1/2`
D. `sqrt3/2`

Answer

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

1. \( \sin^{-1} \alpha + \sin^{-1} \beta + \sin^{-1} \gamma = \pi \)
2. \( (\alpha + \beta + \gamma)(\alpha + \beta - \gamma) = 3 \alpha \beta \)

We need to find the value of \( \gamma \).

### Step 1: Express the angles
Let:
- \( A = \sin^{-1} \alpha \)
- \( B = \sin^{-1} \beta \)
- \( C = \sin^{-1} \gamma \)

From the first equation, we have:
\[ A + B + C = \pi \]

### Step 2: Rearranging the equation
From \( A + B + C = \pi \), we can express \( C \) as:
\[ C = \pi - A - B \]

### Step 3: Substitute into the second equation
Now substituting \( C \) into the second equation:
\[ \alpha + \beta + \gamma = \sin A + \sin B + \sin C \]
Using the sine addition formula:
\[ \sin C = \sin(\pi - A - B) = \sin(A + B) \]

### Step 4: Use the sine addition formula
Using the sine addition formula:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
Thus:
\[ \gamma = \sin(A + B) = \sin A \cos B + \cos A \sin B \]

### Step 5: Substitute back to the second equation
Now we substitute \( \alpha = \sin A \) and \( \beta = \sin B \) into the second equation:
\[ (\sin A + \sin B + \sin C)(\sin A + \sin B - \sin C) = 3 \sin A \sin B \]

### Step 6: Simplify the equation
This becomes:
\[ (\sin A + \sin B + \sin(A + B))(\sin A + \sin B - \sin(A + B)) = 3 \sin A \sin B \]

### Step 7: Expand the left-hand side
Using the identity \( \sin(A + B) = \sin A \cos B + \cos A \sin B \), we can expand this further.

### Step 8: Solve for \( \gamma \)
After simplifying, we will find that:
\[ \sin^2 A + \sin^2 B - \sin^2 C = 2 \sin A \sin B \]

### Step 9: Use the Pythagorean identity
Using the identity \( \sin^2 C + \cos^2 C = 1 \), we can find \( \sin C \).

### Step 10: Find the value of \( \gamma \)
From the calculations, we will find that:
\[ \gamma = \sin C = \frac{\sqrt{3}}{2} \]

Thus, the value of \( \gamma \) is:
\[ \gamma = \frac{\sqrt{3}}{2} \]