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

Answer

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

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

The value of \( \tan^{-1} \left( \frac{-1}{\sqrt{3}} \right) \) corresponds to an angle where the tangent is \( \frac{-1}{\sqrt{3}} \).

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 \( \cos^{-1} \left( \frac{-\sqrt{3}}{2} \right) \)

The value of \( \cos^{-1} \left( \frac{-\sqrt{3}}{2} \right) \) corresponds to an angle where the cosine is \( \frac{-\sqrt{3}}{2} \).

We know that:
\[ \cos \left( \frac{5\pi}{6} \right) = \frac{-\sqrt{3}}{2} \]
Thus,
\[ \cos^{-1} \left( \frac{-\sqrt{3}}{2} \right) = \frac{5\pi}{6} \]

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

The value of \( \sin^{-1} \left( \frac{1}{2} \right) \) corresponds to an angle where the sine is \( \frac{1}{2} \).

We know that:
\[ \sin \left( \frac{\pi}{6} \right) = \frac{1}{2} \]
Thus,
\[ \sin^{-1} \left( \frac{1}{2} \right) = \frac{\pi}{6} \]

### Step 4: Combine the results

Now, we can combine all the evaluated results:
\[ \tan^{-1} \left( \frac{-1}{\sqrt{3}} \right) + \cos^{-1} \left( \frac{-\sqrt{3}}{2} \right) + \sin^{-1} \left( \frac{1}{2} \right) = -\frac{\pi}{6} + \frac{5\pi}{6} + \frac{\pi}{6} \]

### Step 5: Simplify the expression

Combining these:
\[ -\frac{\pi}{6} + \frac{5\pi}{6} + \frac{\pi}{6} = \frac{-\pi + 5\pi + \pi}{6} = \frac{5\pi}{6} \]

### Final Answer

Thus, the value of the expression is:
\[ \frac{5\pi}{6} \]