Find the value `tan (pi/5)+2tan((2pi)/5)+4cot((4pi)/5).`

Answer

To find the value of the expression \( \tan\left(\frac{\pi}{5}\right) + 2\tan\left(\frac{2\pi}{5}\right) + 4\cot\left(\frac{4\pi}{5}\right) \), we can follow these steps:

### Step 1: Identify the angles
Let \( \theta = \frac{\pi}{5} \). Therefore, we have:
- \( \tan\left(\frac{\pi}{5}\right) = \tan(\theta) \)
- \( \tan\left(\frac{2\pi}{5}\right) = \tan(2\theta) \)
- \( \cot\left(\frac{4\pi}{5}\right) = \cot(\pi - \theta) = -\tan(\theta) \)

### Step 2: Calculate \( \tan(\theta) \)
Using the known value:
\[ \tan\left(\frac{\pi}{5}\right) = \tan(36^\circ) = \sqrt{5} - 2 \]

### Step 3: Calculate \( \tan(2\theta) \)
Using the double angle formula:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
Substituting \( \tan(\theta) = \sqrt{5} - 2 \):
\[ \tan(2\theta) = \frac{2(\sqrt{5} - 2)}{1 - (\sqrt{5} - 2)^2} \]
Calculating \( (\sqrt{5} - 2)^2 \):
\[ (\sqrt{5} - 2)^2 = 5 - 4\sqrt{5} + 4 = 9 - 4\sqrt{5} \]
Thus,
\[ 1 - (\sqrt{5} - 2)^2 = 1 - (9 - 4\sqrt{5}) = 4\sqrt{5} - 8 \]
Now substituting back:
\[ \tan(2\theta) = \frac{2(\sqrt{5} - 2)}{4\sqrt{5} - 8} = \frac{\sqrt{5} - 2}{2(\sqrt{5} - 2)} = \frac{1}{2} \]

### Step 4: Calculate \( \cot(4\theta) \)
Since \( \cot(4\theta) = -\tan(\theta) \):
\[ \cot\left(\frac{4\pi}{5}\right) = -\tan\left(\frac{\pi}{5}\right) = -(\sqrt{5} - 2) \]

### Step 5: Substitute values into the expression
Now substituting all values into the original expression:
\[ \tan\left(\frac{\pi}{5}\right) + 2\tan\left(\frac{2\pi}{5}\right) + 4\cot\left(\frac{4\pi}{5}\right) \]
This becomes:
\[ (\sqrt{5} - 2) + 2\left(\frac{1}{2}\right) + 4(-(\sqrt{5} - 2)) \]
Calculating each term:
\[ = (\sqrt{5} - 2) + 1 - 4(\sqrt{5} - 2) \]
\[ = \sqrt{5} - 2 + 1 - 4\sqrt{5} + 8 \]
Combining like terms:
\[ = (\sqrt{5} - 4\sqrt{5}) + (-2 + 1 + 8) \]
\[ = -3\sqrt{5} + 7 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{7 - 3\sqrt{5}} \]