The value of `tan(1/2 cos^(-1)'2/(sqrt(5)))` is

Options

A. `2+sqrt(5)`
B. `sqrt(5) -2`
C. `(sqrt(5)+2)/(2)`
D. `5+sqrt(2)`

Explanation

To find the value of \( \tan\left(\frac{1}{2} \cos^{-1}\left(\frac{2}{\sqrt{5}}\right)\right) \), we will follow these steps:

### Step 1: Let \( \theta = \cos^{-1}\left(\frac{2}{\sqrt{5}}\right) \)
This means that \( \cos(\theta) = \frac{2}{\sqrt{5}} \).

### Step 2: Use the double angle identity for cosine
From the identity \( \cos(2\theta) = 2\cos^2(\theta) - 1 \), we can express \( \cos(2\theta) \):
\[ \cos(2\theta) = 2\left(\frac{2}{\sqrt{5}}\right)^2 - 1 = 2 \cdot \frac{4}{5} - 1 = \frac{8}{5} - 1 = \frac{3}{5} \]

### Step 3: Find \( \sin(2\theta) \)
Using the Pythagorean identity, we can find \( \sin(2\theta) \):
\[ \sin^2(2\theta) + \cos^2(2\theta) = 1 \]
\[ \sin^2(2\theta) + \left(\frac{3}{5}\right)^2 = 1 \]
\[ \sin^2(2\theta) + \frac{9}{25} = 1 \]
\[ \sin^2(2\theta) = 1 - \frac{9}{25} = \frac{16}{25} \]
\[ \sin(2\theta) = \frac{4}{5} \quad (\text{since } 2\theta \text{ is in the first quadrant}) \]

### Step 4: Find \( \tan(\theta) \)
Using the identities for sine and cosine:
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
We need to find \( \sin(\theta) \):
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
\[ \sin^2(\theta) + \left(\frac{2}{\sqrt{5}}\right)^2 = 1 \]
\[ \sin^2(\theta) + \frac{4}{5} = 1 \]
\[ \sin^2(\theta) = 1 - \frac{4}{5} = \frac{1}{5} \]
\[ \sin(\theta) = \frac{1}{\sqrt{5}} \]

Now, we can find \( \tan(\theta) \):
\[ \tan(\theta) = \frac{\frac{1}{\sqrt{5}}}{\frac{2}{\sqrt{5}}} = \frac{1}{2} \]

### Step 5: Find \( \tan\left(\frac{1}{2} \cos^{-1}\left(\frac{2}{\sqrt{5}}\right)\right) \)
Using the half-angle formula:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{\sin(\theta)} \]
Substituting the values:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{1 - \frac{2}{\sqrt{5}}}{\frac{1}{\sqrt{5}}} \]
\[ = \frac{\sqrt{5} - 2}{1} = \sqrt{5} - 2 \]

Thus, the value of \( \tan\left(\frac{1}{2} \cos^{-1}\left(\frac{2}{\sqrt{5}}\right)\right) \) is \( \sqrt{5} - 2 \).

### Final Answer:
\[ \boxed{\sqrt{5} - 2} \]