The vlaue of
`tan^(-1)sqrt(3)-sec^(-1)(-2)+cosec^(-1)(2)/sqrt(3)` is

Options

A. `(pi)/(3)`
B. `-(pi)/(3)`
C. 0
D. `(4pi)/(3)`

Explanation

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

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

### Step 2: Evaluate \( \sec^{-1}(-2) \)
The secant function is defined as:
\[ \sec(x) = \frac{1}{\cos(x)} \]
To find \( \sec^{-1}(-2) \), we need to find an angle \( x \) such that:
\[ \sec(x) = -2 \implies \cos(x) = -\frac{1}{2} \]
The angle \( x \) that satisfies this is:
\[ x = \frac{2\pi}{3} \quad \text{(since } \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\text{)} \]
Thus,
\[ \sec^{-1}(-2) = \frac{2\pi}{3} \]

### Step 3: Evaluate \( \csc^{-1}\left(\frac{2}{\sqrt{3}}\right) \)
We know that:
\[ \csc(x) = \frac{1}{\sin(x)} \]
To find \( \csc^{-1}\left(\frac{2}{\sqrt{3}}\right) \), we need to find an angle \( x \) such that:
\[ \csc(x) = \frac{2}{\sqrt{3}} \implies \sin(x) = \frac{\sqrt{3}}{2} \]
The angle \( x \) that satisfies this is:
\[ x = \frac{\pi}{3} \quad \text{(since } \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\text{)} \]
Thus,
\[ \csc^{-1}\left(\frac{2}{\sqrt{3}}\right) = \frac{\pi}{3} \]

### Step 4: Combine all the values
Now we can substitute back into the original expression:
\[ \tan^{-1}(\sqrt{3}) - \sec^{-1}(-2) + \csc^{-1}\left(\frac{2}{\sqrt{3}}\right) = \frac{\pi}{3} - \frac{2\pi}{3} + \frac{\pi}{3} \]
Calculating this gives:
\[ = \frac{\pi}{3} - \frac{2\pi}{3} + \frac{\pi}{3} = 0 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{0} \]