`tan100^0+tan125^0+tan100^0tan125^0` is equal to

Options

A. 0
B. `1/2`
C. `-1`
D. 1

Explanation

To solve the expression \( \tan 100^\circ + \tan 125^\circ + \tan 100^\circ \tan 125^\circ \), we can use the tangent addition formula.

### Step-by-Step Solution:

1. **Identify the angles**: We have two angles, \( a = 100^\circ \) and \( b = 125^\circ \).

2. **Use the tangent addition formula**: The formula for the tangent of the sum of two angles is given by:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
In our case, we can rewrite the expression as:
\[ \tan 100^\circ + \tan 125^\circ + \tan 100^\circ \tan 125^\circ = \tan 100^\circ + \tan 125^\circ + \tan 100^\circ \tan 125^\circ = \tan 225^\circ \]

3. **Calculate \( \tan 225^\circ \)**: The angle \( 225^\circ \) can be expressed as \( 180^\circ + 45^\circ \). The tangent of \( 225^\circ \) is:
\[ \tan 225^\circ = \tan(180^\circ + 45^\circ) = \tan 45^\circ = 1 \]

4. **Set up the equation**: According to the tangent addition formula:
\[ \tan 225^\circ = \frac{\tan 100^\circ + \tan 125^\circ}{1 - \tan 100^\circ \tan 125^\circ} \]
Since we know \( \tan 225^\circ = 1 \), we can set up the equation:
\[ 1 = \frac{\tan 100^\circ + \tan 125^\circ}{1 - \tan 100^\circ \tan 125^\circ} \]

5. **Cross-multiply**: This gives us:
\[ 1 - \tan 100^\circ \tan 125^\circ = \tan 100^\circ + \tan 125^\circ \]

6. **Rearranging the equation**: Moving \( \tan 100^\circ + \tan 125^\circ \) to the left side, we have:
\[ \tan 100^\circ + \tan 125^\circ + \tan 100^\circ \tan 125^\circ = 1 \]

7. **Conclusion**: Therefore, the value of \( \tan 100^\circ + \tan 125^\circ + \tan 100^\circ \tan 125^\circ \) is equal to:
\[ \boxed{1} \]