The algebraic expression for `f(x)=tan(sin^(-1)(cos("tan"^(-1)(x)/(2))))` is

Options

A. `(2)/(x)`
B. `(x)/(2)`
C. `(1)/(x)`
D. `(2)/(|x|)`

Explanation

To find the algebraic expression for the function \( f(x) = \tan\left(\sin^{-1}\left(\cos\left(\tan^{-1}\left(\frac{x}{2}\right)\right)\right)\right) \), we will follow these steps:

### Step 1: Set up the problem
We start with the function:
\[ f(x) = \tan\left(\sin^{-1}\left(\cos\left(\tan^{-1}\left(\frac{x}{2}\right)\right)\right)\right) \]

### Step 2: Let \( \theta = \tan^{-1}\left(\frac{x}{2}\right) \)
From this, we can express \( \tan \theta \):
\[ \tan \theta = \frac{x}{2} \]
Using the identity \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \), we can consider a right triangle where the opposite side is \( x \) and the adjacent side is \( 2 \).

### Step 3: Find the hypotenuse
Using the Pythagorean theorem, we find the hypotenuse \( h \):
\[ h = \sqrt{x^2 + 2^2} = \sqrt{x^2 + 4} \]

### Step 4: Find \( \cos \theta \)
Now, we can find \( \cos \theta \):
\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{2}{\sqrt{x^2 + 4}} \]

### Step 5: Substitute into \( f(x) \)
Now we substitute \( \cos \theta \) into the function:
\[ f(x) = \tan\left(\sin^{-1}\left(\frac{2}{\sqrt{x^2 + 4}}\right)\right) \]

### Step 6: Let \( \phi = \sin^{-1}\left(\frac{2}{\sqrt{x^2 + 4}}\right) \)
From this, we can express \( \sin \phi \):
\[ \sin \phi = \frac{2}{\sqrt{x^2 + 4}} \]

### Step 7: Find \( \cos \phi \)
Using the Pythagorean identity:
\[ \cos \phi = \sqrt{1 - \sin^2 \phi} = \sqrt{1 - \left(\frac{2}{\sqrt{x^2 + 4}}\right)^2} = \sqrt{1 - \frac{4}{x^2 + 4}} = \sqrt{\frac{x^2}{x^2 + 4}} = \frac{|x|}{\sqrt{x^2 + 4}} \]

### Step 8: Find \( \tan \phi \)
Now we can find \( \tan \phi \):
\[ \tan \phi = \frac{\sin \phi}{\cos \phi} = \frac{\frac{2}{\sqrt{x^2 + 4}}}{\frac{|x|}{\sqrt{x^2 + 4}}} = \frac{2}{|x|} \]

### Step 9: Final expression for \( f(x) \)
Thus, the algebraic expression for \( f(x) \) is:
\[ f(x) = \frac{2}{|x|} \]

### Summary
The final answer is:
\[ f(x) = \frac{2}{|x|} \]