Find the exact value of the expression `tan(pi/20)-tan((3pi)/20)+tan((5pi)/20)-tan((7pi)/20)+tan((9pi)/20).`

Answer

To find the exact value of the expression

\[ E = \tan\left(\frac{\pi}{20}\right) - \tan\left(\frac{3\pi}{20}\right) + \tan\left(\frac{5\pi}{20}\right) - \tan\left(\frac{7\pi}{20}\right) + \tan\left(\frac{9\pi}{20}\right), \]

we will follow these steps:

### Step 1: Convert radians to degrees
We know that \(\pi\) radians is equivalent to \(180^\circ\). Therefore, we can convert each term:

\[ E = \tan\left(9^\circ\right) - \tan\left(27^\circ\right) + \tan\left(45^\circ\right) - \tan\left(63^\circ\right) + \tan\left(81^\circ\right). \]

### Step 2: Use the identity \(\tan(90^\circ - x) = \cot(x)\)
We can simplify \(\tan(63^\circ)\) and \(\tan(81^\circ)\):

\[ \tan(63^\circ) = \cot(27^\circ) \quad \text{and} \quad \tan(81^\circ) = \cot(9^\circ). \]

Thus, we can rewrite \(E\):

\[ E = \tan(9^\circ) - \tan(27^\circ) + \tan(45^\circ) - \cot(27^\circ) + \cot(9^\circ). \]

### Step 3: Substitute \(\tan(45^\circ)\)
Since \(\tan(45^\circ) = 1\), we can substitute this value into our expression:

\[ E = \tan(9^\circ) - \tan(27^\circ) + 1 - \cot(27^\circ) + \cot(9^\circ). \]

### Step 4: Combine terms
Now, we can group the terms involving \(\tan\) and \(\cot\):

\[ E = \left(\tan(9^\circ) + \cot(9^\circ)\right) - \left(\tan(27^\circ) + \cot(27^\circ)\right) + 1. \]

### Step 5: Use the identity \(\tan(x) + \cot(x) = \frac{\sin^2(x) + \cos^2(x)}{\sin(x)\cos(x)} = \frac{1}{\sin(x)\cos(x)}\)
This means:

\[ \tan(x) + \cot(x) = \frac{1}{\sin(x)\cos(x)} = \frac{2}{\sin(2x)}. \]

Thus, we have:

\[ E = \frac{2}{\sin(18^\circ)} - \frac{2}{\sin(54^\circ)} + 1. \]

### Step 6: Use known values of \(\sin(18^\circ)\) and \(\sin(54^\circ)\)
We know:

\[ \sin(18^\circ) = \frac{\sqrt{5}-1}{4} \quad \text{and} \quad \sin(54^\circ) = \frac{\sqrt{5}+1}{4}. \]

Substituting these values gives:

\[ E = \frac{2}{\frac{\sqrt{5}-1}{4}} - \frac{2}{\frac{\sqrt{5}+1}{4}} + 1. \]

### Step 7: Simplify the expression
This simplifies to:

\[ E = \frac{8}{\sqrt{5}-1} - \frac{8}{\sqrt{5}+1} + 1. \]

Finding a common denominator:

\[ E = \frac{8(\sqrt{5}+1) - 8(\sqrt{5}-1)}{(\sqrt{5}-1)(\sqrt{5}+1)} + 1. \]

This results in:

\[ E = \frac{8\sqrt{5} + 8 - 8\sqrt{5} + 8}{4} + 1 = \frac{16}{4} + 1 = 4 + 1 = 5. \]

### Final Answer
Thus, the exact value of the expression is

\[ \boxed{5}. \]