The value of `tan[(1)/(2)cos^(-1)((sqrt5)/(3))]` is

Options

A. `(3+sqrt5)/(2)`
B. `(3-sqrt5)/(2)`
C. `(-3+sqrt5)/(2)`
D. `(-3-sqrt5)/(2)`

Answer

To find the value of \( \tan\left(\frac{1}{2} \cos^{-1}\left(\frac{\sqrt{5}}{3}\right)\right) \), we can follow these steps:

### Step 1: Define the angle
Let \( \theta = \cos^{-1}\left(\frac{\sqrt{5}}{3}\right) \). This implies that \( \cos \theta = \frac{\sqrt{5}}{3} \).

### Step 2: Use the half-angle formula for tangent
We can use the half-angle formula for tangent:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{\sin \theta}{1 + \cos \theta} \]

### Step 3: Find \( \sin \theta \)
To find \( \sin \theta \), we can use the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \cos \theta \):
\[ \sin^2 \theta + \left(\frac{\sqrt{5}}{3}\right)^2 = 1 \]
\[ \sin^2 \theta + \frac{5}{9} = 1 \]
\[ \sin^2 \theta = 1 - \frac{5}{9} = \frac{4}{9} \]
Taking the square root:
\[ \sin \theta = \frac{2}{3} \quad (\text{since } \theta \text{ is in the range } [0, \pi]) \]

### Step 4: Substitute \( \sin \theta \) and \( \cos \theta \) into the half-angle formula
Now we can substitute \( \sin \theta \) and \( \cos \theta \) into the half-angle formula:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{\sin \theta}{1 + \cos \theta} = \frac{\frac{2}{3}}{1 + \frac{\sqrt{5}}{3}} \]

### Step 5: Simplify the expression
First, simplify the denominator:
\[ 1 + \frac{\sqrt{5}}{3} = \frac{3 + \sqrt{5}}{3} \]
Now substituting back:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{\frac{2}{3}}{\frac{3 + \sqrt{5}}{3}} = \frac{2}{3 + \sqrt{5}} \]

### Step 6: Rationalize the denominator
To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{2(3 - \sqrt{5})}{(3 + \sqrt{5})(3 - \sqrt{5})} \]
Calculating the denominator:
\[ (3 + \sqrt{5})(3 - \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4 \]
So we have:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{2(3 - \sqrt{5})}{4} = \frac{3 - \sqrt{5}}{2} \]

### Final Answer
Thus, the value of \( \tan\left(\frac{1}{2} \cos^{-1}\left(\frac{\sqrt{5}}{3}\right)\right) \) is:
\[ \frac{3 - \sqrt{5}}{2} \]

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