`tan""(2pi)/(5)-tan""(pi)/(15)-sqrt3tan""(2pi)/(5)tan""(pi)/(15)` is equal to

Options

A. `-sqrt(3)`
B. `(1)/sqrt(3)`
C. 1
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 can follow these steps:

### Step 1: Simplify the angles
First, we need to simplify the angles involved. We can find a common angle for the tangent function.

We know:
\[ \frac{2\pi}{5} - \frac{\pi}{15} \]
To simplify this, we need a common denominator. The least common multiple of 5 and 15 is 15. Thus, we convert:
\[ \frac{2\pi}{5} = \frac{6\pi}{15} \]
So,
\[ \frac{2\pi}{5} - \frac{\pi}{15} = \frac{6\pi}{15} - \frac{\pi}{15} = \frac{5\pi}{15} = \frac{\pi}{3} \]

### Step 2: Use the tangent subtraction formula
Now we can use the tangent subtraction formula:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
Let \( A = \frac{2\pi}{5} \) and \( B = \frac{\pi}{15} \). Then we have:
\[ \tan\left(\frac{2\pi}{5} - \frac{\pi}{15}\right) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]

### Step 3: Set up the equation
From the tangent subtraction formula, we can set up the equation:
\[ \tan\left(\frac{2\pi}{5}\right) - \tan\left(\frac{\pi}{15}\right) = \sqrt{3} \left(1 + \tan\left(\frac{2\pi}{5}\right) \tan\left(\frac{\pi}{15}\right)\right) \]

### Step 4: Rearranging the equation
Rearranging gives us:
\[ \tan\left(\frac{2\pi}{5}\right) - \tan\left(\frac{\pi}{15}\right) - \sqrt{3} \tan\left(\frac{2\pi}{5}\right) \tan\left(\frac{\pi}{15}\right) = \sqrt{3} \]

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

Therefore, the final answer is:
\[ \sqrt{3} \]