If `tan(A+B)=(1)/(2),tan(A-B)=(1)/(3)`, then find the value of `tan2A`.

Options

A. 5
B. 7
C. 1
D. 3

Answer

To find the value of \( \tan 2A \) given that \( \tan(A+B) = \frac{1}{2} \) and \( \tan(A-B) = \frac{1}{3} \), we can use the tangent addition and subtraction formulas.

### Step 1: Use the tangent addition and subtraction formulas
We know that:
\[ \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
and
\[ \tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]

Let \( \tan A = x \) and \( \tan B = y \). Then we can rewrite the equations as:
\[ \frac{x + y}{1 - xy} = \frac{1}{2} \quad (1) \]
\[ \frac{x - y}{1 + xy} = \frac{1}{3} \quad (2) \]

### Step 2: Cross-multiply the equations
From equation (1):
\[ x + y = \frac{1}{2}(1 - xy) \implies 2(x + y) = 1 - xy \implies 2x + 2y + xy = 1 \quad (3) \]

From equation (2):
\[ x - y = \frac{1}{3}(1 + xy) \implies 3(x - y) = 1 + xy \implies 3x - 3y - xy = 1 \quad (4) \]

### Step 3: Solve the system of equations
Now we have two equations (3) and (4):
1. \( 2x + 2y + xy = 1 \)
2. \( 3x - 3y - xy = 1 \)

From equation (3), we can express \( xy \):
\[ xy = 1 - 2x - 2y \quad (5) \]

Substituting (5) into equation (4):
\[ 3x - 3y - (1 - 2x - 2y) = 1 \]
\[ 3x - 3y - 1 + 2x + 2y = 1 \]
\[ 5x - y - 2 = 1 \]
\[ 5x - y = 3 \quad (6) \]

### Step 4: Express \( y \) in terms of \( x \)
From equation (6):
\[ y = 5x - 3 \quad (7) \]

### Step 5: Substitute \( y \) back into equation (5)
Substituting (7) into (5):
\[ x(5x - 3) = 1 - 2x - 2(5x - 3) \]
\[ 5x^2 - 3x = 1 - 2x - 10x + 6 \]
\[ 5x^2 - 3x = 7 - 12x \]
\[ 5x^2 + 9x - 7 = 0 \]

### Step 6: Solve the quadratic equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-9 \pm \sqrt{9^2 - 4 \cdot 5 \cdot (-7)}}{2 \cdot 5} \]
\[ x = \frac{-9 \pm \sqrt{81 + 140}}{10} \]
\[ x = \frac{-9 \pm \sqrt{221}}{10} \]

### Step 7: Find \( \tan 2A \)
Using the formula:
\[ \tan 2A = \frac{2\tan A}{1 - \tan^2 A} \]
Substituting \( x \) into this formula will give us the value of \( \tan 2A \).

### Final Calculation
Now, we can calculate \( \tan 2A \) using the values of \( x \) we found.