Find the value of
`sin^(-1)(-(sqrt(3))/(2))+cos^(-1)((1)/(2))+tan^(-1)(-(1)/(sqrt(3)))`

Answer

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

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

Using the property of inverse sine, we know that:
\[ \sin^{-1}(-x) = -\sin^{-1}(x) \]
Thus,
\[ \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \]
Now, we find \( \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \). The angle whose sine is \( \frac{\sqrt{3}}{2} \) is \( \frac{\pi}{3} \):
\[ \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} \]
Therefore,
\[ \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3} \]

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

The angle whose cosine is \( \frac{1}{2} \) is \( \frac{\pi}{3} \):
\[ \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \]

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

Using the property of inverse tangent, we have:
\[ \tan^{-1}(-x) = -\tan^{-1}(x) \]
Thus,
\[ \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) = -\tan^{-1}\left(\frac{1}{\sqrt{3}}\right) \]
Now, we find \( \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) \). The angle whose tangent is \( \frac{1}{\sqrt{3}} \) is \( \frac{\pi}{6} \):
\[ \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \]
Therefore,
\[ \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{6} \]

### Step 4: Combine all the values

Now we can combine all the evaluated terms:
\[ \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) + \cos^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{3} + \frac{\pi}{3} - \frac{\pi}{6} \]
The first two terms cancel each other:
\[ -\frac{\pi}{3} + \frac{\pi}{3} = 0 \]
Thus, we have:
\[ 0 - \frac{\pi}{6} = -\frac{\pi}{6} \]

### Final Answer
The value of \( \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) + \cos^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) \) is:
\[ \boxed{-\frac{\pi}{6}} \]