Two sides of a triangle are `(sqrt(3)+1)` and `(sqrt(3)-1)`, and the angle between them is `60^(@)`. The difference of the remaining angles is

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `90^(@)`

Answer

To find the difference of the remaining angles in the triangle with sides \( A = \sqrt{3} + 1 \), \( B = \sqrt{3} - 1 \), and the angle \( C = 60^\circ \) between them, we can use the formula for the tangent of the half-angle difference:

\[ \tan\left(\frac{A - B}{2}\right) = \frac{A - B}{A + B} \cdot \cot\left(\frac{C}{2}\right) \]

### Step 1: Calculate \( A - B \) and \( A + B \)

First, we calculate:

\[ A - B = (\sqrt{3} + 1) - (\sqrt{3} - 1) = 2 \]

\[ A + B = (\sqrt{3} + 1) + (\sqrt{3} - 1) = 2\sqrt{3} \]

### Step 2: Calculate \( \cot\left(\frac{C}{2}\right) \)

Since \( C = 60^\circ \):

\[ \frac{C}{2} = 30^\circ \]

We know:

\[ \cot(30^\circ) = \frac{1}{\tan(30^\circ)} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \]

### Step 3: Substitute values into the tangent formula

Now we substitute the values into the tangent formula:

\[ \tan\left(\frac{A - B}{2}\right) = \frac{2}{2\sqrt{3}} \cdot \sqrt{3} \]

### Step 4: Simplify the expression

Simplifying gives:

\[ \tan\left(\frac{A - B}{2}\right) = \frac{2\sqrt{3}}{2\sqrt{3}} = 1 \]

### Step 5: Find \( \frac{A - B}{2} \)

Since \( \tan\left(\frac{A - B}{2}\right) = 1 \), we have:

\[ \frac{A - B}{2} = 45^\circ \]

Thus,

\[ A - B = 90^\circ \]

### Step 6: Find the difference of the remaining angles

The remaining angles \( A \) and \( B \) in the triangle must sum up to \( 120^\circ \) (since the sum of angles in a triangle is \( 180^\circ \) and we already have \( C = 60^\circ \)). Therefore, we can express the angles as:

\[ A + B = 120^\circ \]

Now, using \( A - B = 90^\circ \) and \( A + B = 120^\circ \):

Adding these two equations:

\[ 2A = 210^\circ \implies A = 105^\circ \]

Subtracting the second from the first:

\[ 2B = 30^\circ \implies B = 15^\circ \]

### Step 7: Calculate the difference of the remaining angles

Now, the difference of the remaining angles \( A \) and \( B \) is:

\[ A - B = 105^\circ - 15^\circ = 90^\circ \]

### Final Answer

The difference of the remaining angles is \( 90^\circ \).

---