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

Options

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

Answer

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

### Step 1: Identify the angles
We have:
- \( a = 100^\circ \)
- \( b = 125^\circ \)

### Step 2: Calculate the sum of the angles
Calculate \( a + b \):
\[ a + b = 100^\circ + 125^\circ = 225^\circ \]

### Step 3: Use the tangent addition formula
The tangent addition formula states:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
Substituting our values:
\[ \tan(225^\circ) = \frac{\tan(100^\circ) + \tan(125^\circ)}{1 - \tan(100^\circ) \tan(125^\circ)} \]

### Step 4: Find the value of \( \tan(225^\circ) \)
We know that:
\[ \tan(225^\circ) = \tan(180^\circ + 45^\circ) = \tan(45^\circ) = 1 \]
Thus:
\[ 1 = \frac{\tan(100^\circ) + \tan(125^\circ)}{1 - \tan(100^\circ) \tan(125^\circ)} \]

### Step 5: Rearranging the equation
Multiply both sides by \( 1 - \tan(100^\circ) \tan(125^\circ) \):
\[ 1 - \tan(100^\circ) \tan(125^\circ) = \tan(100^\circ) + \tan(125^\circ) \]

### Step 6: Add \( \tan(100^\circ) \tan(125^\circ) \) to both sides
Rearranging gives us:
\[ \tan(100^\circ) + \tan(125^\circ) + \tan(100^\circ) \tan(125^\circ) = 1 \]

### Final Answer
Thus, the value of the expression \( \tan(100^\circ) + \tan(125^\circ) + \tan(100^\circ) \tan(125^\circ) \) is:
\[ \boxed{1} \]