`tan(100^(@))+tan(125^(@))+tan(100^(@))tan(125^(@))=`

Options

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

Answer

To solve the equation \( \tan(100^\circ) + \tan(125^\circ) + \tan(100^\circ) \tan(125^\circ) \), we can use the formula for the tangent of the sum of two angles.

### Step-by-step Solution:

1. **Identify the Angles**:
Let \( A = 100^\circ \) and \( B = 125^\circ \).

2. **Use the Tangent Addition Formula**:
The formula for the tangent of the sum of two angles is given by:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Therefore, we can express \( \tan(100^\circ + 125^\circ) \) as:
\[ \tan(225^\circ) = \frac{\tan(100^\circ) + \tan(125^\circ)}{1 - \tan(100^\circ) \tan(125^\circ)} \]

3. **Calculate \( \tan(225^\circ) \)**:
We know that:
\[ \tan(225^\circ) = \tan(180^\circ + 45^\circ) = \tan(45^\circ) = 1 \]

4. **Set Up the Equation**:
From the tangent addition formula, we have:
\[ 1 = \frac{\tan(100^\circ) + \tan(125^\circ)}{1 - \tan(100^\circ) \tan(125^\circ)} \]

5. **Cross Multiply**:
Cross multiplying gives us:
\[ 1 - \tan(100^\circ) \tan(125^\circ) = \tan(100^\circ) + \tan(125^\circ) \]

6. **Rearranging the Equation**:
Rearranging the equation, we can write:
\[ \tan(100^\circ) + \tan(125^\circ) + \tan(100^\circ) \tan(125^\circ) = 1 \]

### Final Answer:
Thus, we conclude that:
\[ \tan(100^\circ) + \tan(125^\circ) + \tan(100^\circ) \tan(125^\circ) = 1 \]