`tan((1)/(2).cos^(-1).(2)/(sqrt(5)))=`

Options

A. `5-sqrt(2)`
B. `sqrt(5)-2`
C. `5+sqrt(2)`
D. none of these

Answer

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

### Step 1: Let \( x = \cos^{-1}\left(\frac{2}{\sqrt{5}}\right) \)
By defining \( x \) in this way, we can express \( \tan\left(\frac{1}{2} x\right) \) in terms of \( \cos x \).

### Step 2: Use the identity for \( \tan\left(\frac{x}{2}\right) \)
The identity for \( \tan\left(\frac{x}{2}\right) \) is given by:
\[ \tan\left(\frac{x}{2}\right) = \sqrt{\frac{1 - \cos x}{1 + \cos x}} \]
Substituting \( \cos x = \frac{2}{\sqrt{5}} \) into the formula gives:
\[ \tan\left(\frac{x}{2}\right) = \sqrt{\frac{1 - \frac{2}{\sqrt{5}}}{1 + \frac{2}{\sqrt{5}}}} \]

### Step 3: Simplify the expression
Now we simplify the numerator and denominator:
- **Numerator**:
\[ 1 - \frac{2}{\sqrt{5}} = \frac{\sqrt{5} - 2}{\sqrt{5}} \]
- **Denominator**:
\[ 1 + \frac{2}{\sqrt{5}} = \frac{\sqrt{5} + 2}{\sqrt{5}} \]

### Step 4: Substitute back into the formula
Now substituting these back into the formula:
\[ \tan\left(\frac{x}{2}\right) = \sqrt{\frac{\frac{\sqrt{5} - 2}{\sqrt{5}}}{\frac{\sqrt{5} + 2}{\sqrt{5}}}} = \sqrt{\frac{\sqrt{5} - 2}{\sqrt{5} + 2}} \]

### Step 5: Rationalize the expression
To rationalize the expression, we multiply the numerator and denominator by \( \sqrt{5} - 2 \):
\[ \tan\left(\frac{x}{2}\right) = \sqrt{\frac{(\sqrt{5} - 2)^2}{(\sqrt{5} + 2)(\sqrt{5} - 2)}} \]
Calculating the denominator:
\[ (\sqrt{5} + 2)(\sqrt{5} - 2) = 5 - 4 = 1 \]
Thus, we have:
\[ \tan\left(\frac{x}{2}\right) = \sqrt{(\sqrt{5} - 2)^2} = \sqrt{5} - 2 \]

### Final Answer
Thus, the value of \( \tan\left(\frac{1}{2} \cos^{-1}\left(\frac{2}{\sqrt{5}}\right)\right) \) is:
\[ \sqrt{5} - 2 \]