What is the value of the following functions (using principal value)
`tan^(-1)((1)/(sqrt(3)))-sec^(-1)((2)/(sqrt(3)))`

Answer

To solve the expression \( \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) - \sec^{-1}\left(\frac{2}{\sqrt{3}}\right) \), we will evaluate each term step by step.

### Step 1: Evaluate \( \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) \)

We know that:
\[ \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \]
Thus,
\[ \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \]

### Step 2: Evaluate \( \sec^{-1}\left(\frac{2}{\sqrt{3}}\right) \)

Recall that:
\[ \sec(x) = \frac{1}{\cos(x)} \]
So, we need to find \( x \) such that:
\[ \sec(x) = \frac{2}{\sqrt{3}} \implies \cos(x) = \frac{\sqrt{3}}{2} \]
The angle \( x \) that satisfies this is:
\[ x = \frac{\pi}{6} \]
Thus,
\[ \sec^{-1}\left(\frac{2}{\sqrt{3}}\right) = \frac{\pi}{6} \]

### Step 3: Combine the results

Now we substitute back into the original expression:
\[ \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) - \sec^{-1}\left(\frac{2}{\sqrt{3}}\right) = \frac{\pi}{6} - \frac{\pi}{6} = 0 \]

### Final Answer

The value of the expression \( \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) - \sec^{-1}\left(\frac{2}{\sqrt{3}}\right) \) is:
\[ \boxed{0} \]