If tan (A+B) =1 and tan (A-B)= `(1)/sqrt(3),0^(@)lt A+B lt 90^(@),` then the value of sin (3A-7B) is :

Options

A. `(sqrt(3))/(2)`
B. `(1)/(2)`
C. `(1)/sqrt(2)`
D. 1

Answer

To solve the problem step-by-step, we will start with the given equations and find the values of angles A and B.

### Step 1: Analyze the given equations
We have:
1. \( \tan(A + B) = 1 \)
2. \( \tan(A - B) = \frac{1}{\sqrt{3}} \)

### Step 2: Determine angles from tangent values
From \( \tan(A + B) = 1 \), we know that:
- \( A + B = 45^\circ \) (since \( \tan 45^\circ = 1 \))

From \( \tan(A - B) = \frac{1}{\sqrt{3}} \), we know that:
- \( A - B = 30^\circ \) (since \( \tan 30^\circ = \frac{1}{\sqrt{3}} \))

### Step 3: Set up a system of equations
Now we have two equations:
1. \( A + B = 45^\circ \)
2. \( A - B = 30^\circ \)

### Step 4: Solve for A and B
To find A and B, we can add the two equations:
\[ (A + B) + (A - B) = 45^\circ + 30^\circ \]
This simplifies to:
\[ 2A = 75^\circ \implies A = \frac{75^\circ}{2} = 37.5^\circ \]

Now, substituting \( A \) back into one of the original equations to find \( B \):
\[ A + B = 45^\circ \implies 37.5^\circ + B = 45^\circ \implies B = 45^\circ - 37.5^\circ = 7.5^\circ \]

### Step 5: Calculate \( 3A - 7B \)
Now we need to find \( 3A - 7B \):
\[ 3A = 3 \times 37.5^\circ = 112.5^\circ \]
\[ 7B = 7 \times 7.5^\circ = 52.5^\circ \]
Now, substituting these values:
\[ 3A - 7B = 112.5^\circ - 52.5^\circ = 60^\circ \]

### Step 6: Find \( \sin(3A - 7B) \)
Now we find \( \sin(60^\circ) \):
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]

### Final Answer
Thus, the value of \( \sin(3A - 7B) \) is:
\[ \frac{\sqrt{3}}{2} \]