Find the principal value of `cos^(-1)(sqrt3/2)+cot^(-1)(1/sqrt3)+cosec^(-1)(-2)`

Answer

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

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

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

\[ \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6} \]

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

The value of \( \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) \) corresponds to the angle whose cotangent is \( \frac{1}{\sqrt{3}} \). This angle is:

\[ \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{3} \]

### Step 3: Evaluate \( \csc^{-1}(-2) \)

The value of \( \csc^{-1}(-2) \) corresponds to the angle whose cosecant is \( -2 \). The cosecant function is the reciprocal of the sine function. Therefore, we have:

\[ \csc^{-1}(-2) = -\sin^{-1}\left(-\frac{1}{2}\right) \]

Since \( \sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6} \), we get:

\[ \csc^{-1}(-2) = -\left(-\frac{\pi}{6}\right) = \frac{\pi}{6} \]

### Step 4: Combine the results

Now we combine all the evaluated terms:

\[ \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) + \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) + \csc^{-1}(-2) = \frac{\pi}{6} + \frac{\pi}{3} + \frac{\pi}{6} \]

### Step 5: Simplify the expression

To simplify, we convert \( \frac{\pi}{3} \) into sixths:

\[ \frac{\pi}{3} = \frac{2\pi}{6} \]

Now, substituting back, we have:

\[ \frac{\pi}{6} + \frac{2\pi}{6} + \frac{\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3} \]

### Final Answer

Thus, the principal value of the expression is:

\[ \frac{2\pi}{3} \]

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