`tan""(2pi)/(5)-tan""(pi)/(15)-sqrt(3)tan""(2pi)/(5)tan""(pi)/(15)=?`

Options

A. 0
B. 1
C. `sqrt(2)`
D. `sqrt(3)`

Answer

To solve the expression \( \tan\left(\frac{2\pi}{5}\right) - \tan\left(\frac{\pi}{15}\right) - \sqrt{3} \tan\left(\frac{2\pi}{5}\right) \tan\left(\frac{\pi}{15}\right) \), we will follow these steps:

### Step 1: Calculate \( \tan\left(\frac{2\pi}{5}\right) \)
First, we need to find the value of \( \tan\left(\frac{2\pi}{5}\right) \).

\[ \frac{2\pi}{5} = \frac{2 \times 180}{5} = 72^\circ \]
Thus,
\[ \tan\left(\frac{2\pi}{5}\right) = \tan(72^\circ) \]

### Step 2: Calculate \( \tan\left(\frac{\pi}{15}\right) \)
Next, we calculate \( \tan\left(\frac{\pi}{15}\right) \).

\[ \frac{\pi}{15} = \frac{180}{15} = 12^\circ \]
Thus,
\[ \tan\left(\frac{\pi}{15}\right) = \tan(12^\circ) \]

### Step 3: Use the tangent subtraction formula
We can use the tangent subtraction formula:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
Here, we can let \( A = 72^\circ \) and \( B = 12^\circ \).

### Step 4: Substitute into the formula
Using the values we have:
\[ \tan(72^\circ - 12^\circ) = \tan(60^\circ) = \sqrt{3} \]
So we have:
\[ \sqrt{3} = \frac{\tan(72^\circ) - \tan(12^\circ)}{1 + \tan(72^\circ) \tan(12^\circ)} \]

### Step 5: Rearranging the equation
Cross-multiplying gives us:
\[ \sqrt{3}(1 + \tan(72^\circ) \tan(12^\circ) ) = \tan(72^\circ) - \tan(12^\circ) \]
Rearranging this gives:
\[ \tan(72^\circ) - \tan(12^\circ) - \sqrt{3} \tan(72^\circ) \tan(12^\circ) = 0 \]

### Step 6: Final expression
Thus, we can conclude that:
\[ \tan\left(\frac{2\pi}{5}\right) - \tan\left(\frac{\pi}{15}\right) - \sqrt{3} \tan\left(\frac{2\pi}{5}\right) \tan\left(\frac{\pi}{15}\right) = 0 \]

### Final Answer
The final answer is:
\[ 0 \]