If `tan (A - B) = (1)/( sqrt3) and tan (A + B) = sqrt3. ` then the values of A and B are

Options

A. `45 ^(@) , 15 ^(@)`
B. `40^(@) , 20^(@)`
C. `30^(@), 60^(@)`
D. None of these

Answer

To solve the problem, we have the equations:

1. \( \tan(A - B) = \frac{1}{\sqrt{3}} \)
2. \( \tan(A + B) = \sqrt{3} \)

### Step-by-Step Solution:

**Step 1: Identify the angles corresponding to the tangents.**

From the first equation, we know that:

\[ \tan(A - B) = \frac{1}{\sqrt{3}} \implies A - B = 30^\circ \quad \text{(since } \tan 30^\circ = \frac{1}{\sqrt{3}} \text{)} \]

From the second equation, we have:

\[ \tan(A + B) = \sqrt{3} \implies A + B = 60^\circ \quad \text{(since } \tan 60^\circ = \sqrt{3} \text{)} \]

**Step 2: Set up the equations.**

Now we have two equations:

1. \( A - B = 30^\circ \) (Equation 1)
2. \( A + B = 60^\circ \) (Equation 2)

**Step 3: Solve the equations.**

To find the values of \( A \) and \( B \), we can add both equations:

\[ (A - B) + (A + B) = 30^\circ + 60^\circ \]

This simplifies to:

\[ 2A = 90^\circ \]

Dividing both sides by 2:

\[ A = 45^\circ \]

**Step 4: Substitute to find \( B \).**

Now that we have \( A \), we can substitute it back into either equation to find \( B \). Using Equation 1:

\[ 45^\circ - B = 30^\circ \]

Rearranging gives:

\[ B = 45^\circ - 30^\circ = 15^\circ \]

### Final Result:

Thus, the values of \( A \) and \( B \) are:

\[ A = 45^\circ, \quad B = 15^\circ \]

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