Find the value of `tan^(-1)(-1/(sqrt(3)))+cot^(-1)((1)/(sqrt(3))) + tan^(-1)[sin((-pi)/(2))]`.

Explanation

To solve the expression \( \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) + \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) + \tan^{-1}[\sin(-\frac{\pi}{2})] \), 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,
\[ \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \]
Since we have a negative value, we can write:
\[ \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) = -\tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{6} \]

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

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

### Step 3: Evaluate \( \tan^{-1}[\sin(-\frac{\pi}{2})] \)

We know that:
\[ \sin\left(-\frac{\pi}{2}\right) = -1 \]
Thus,
\[ \tan^{-1}(-1) = -\frac{\pi}{4} \]

### Step 4: Combine all the values

Now we can substitute the values we found into the original expression:
\[ \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) + \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) + \tan^{-1}[\sin(-\frac{\pi}{2})] \]
Substituting the values:
\[ -\frac{\pi}{6} + \frac{\pi}{3} - \frac{\pi}{4} \]

### Step 5: Find a common denominator

The least common multiple of the denominators \(6\), \(3\), and \(4\) is \(12\). We will convert each term:
\[ -\frac{\pi}{6} = -\frac{2\pi}{12} \]
\[ \frac{\pi}{3} = \frac{4\pi}{12} \]
\[ -\frac{\pi}{4} = -\frac{3\pi}{12} \]

### Step 6: Combine the fractions

Now we can combine the fractions:
\[ -\frac{2\pi}{12} + \frac{4\pi}{12} - \frac{3\pi}{12} = \frac{-2\pi + 4\pi - 3\pi}{12} = \frac{-1\pi}{12} = -\frac{\pi}{12} \]

### Final Answer

Thus, the value of the expression is:
\[ \boxed{-\frac{\pi}{12}} \]