Find the simplified value of the following expression:
`tan["cos"^(-1)1/2+tan^(-1)(-1/(sqrt(3)))]`

Explanation

To find the simplified value of the expression \( \tan\left(\cos^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right)\right) \), we can follow these steps:

### Step 1: Evaluate \( \cos^{-1}\left(\frac{1}{2}\right) \)
The value of \( \cos^{-1}\left(\frac{1}{2}\right) \) corresponds to the angle whose cosine is \( \frac{1}{2} \). This angle is \( \frac{\pi}{3} \) radians.

### Step 2: Evaluate \( \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) \)
The value of \( \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) \) corresponds to the angle whose tangent is \( -\frac{1}{\sqrt{3}} \). This angle is \( -\frac{\pi}{6} \) radians.

### Step 3: Combine the angles
Now, we can substitute the values we found into the expression:
\[ \tan\left(\cos^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right)\right) = \tan\left(\frac{\pi}{3} - \frac{\pi}{6}\right) \]

### Step 4: Simplify the angle
To simplify \( \frac{\pi}{3} - \frac{\pi}{6} \), we need a common denominator:
\[ \frac{\pi}{3} = \frac{2\pi}{6} \]
Thus,
\[ \frac{\pi}{3} - \frac{\pi}{6} = \frac{2\pi}{6} - \frac{\pi}{6} = \frac{\pi}{6} \]

### Step 5: Evaluate \( \tan\left(\frac{\pi}{6}\right) \)
The value of \( \tan\left(\frac{\pi}{6}\right) \) is \( \frac{1}{\sqrt{3}} \).

### Final Answer
Thus, the simplified value of the expression is:
\[ \frac{1}{\sqrt{3}} \]

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