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

Answer

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

### Step 1: Evaluate \( \cot^{-1}(-\sqrt{3}) \)

The cotangent function is negative in the second and fourth quadrants. We know that:

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

Thus,

\[ \cot\left(\frac{2\pi}{3}\right) = -\sqrt{3} \]

Since the principal value of \( \cot^{-1}(x) \) lies in the interval \( (0, \pi) \), we have:

\[ \cot^{-1}(-\sqrt{3}) = \frac{2\pi}{3} \]

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

The cosecant function is negative in the third and fourth quadrants. We know that:

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

Thus,

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

Now, multiplying by 2 gives:

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

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

The cosine function is negative in the second quadrant. 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 4: Combine the results

Now, we can combine all the results:

\[ \cot^{-1}(-\sqrt{3}) + 2 \csc^{-1}(-2) + \cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) = \frac{2\pi}{3} - \frac{\pi}{3} + \frac{5\pi}{6} \]

Calculating this step by step:

1. Combine \( \frac{2\pi}{3} - \frac{\pi}{3} \):

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

2. Now add \( \frac{5\pi}{6} \):

To add \( \frac{\pi}{3} \) and \( \frac{5\pi}{6} \), we need a common denominator. The least common multiple of 3 and 6 is 6.

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

Now we can add:

\[ \frac{2\pi}{6} + \frac{5\pi}{6} = \frac{2\pi + 5\pi}{6} = \frac{7\pi}{6} \]

### Final Answer:

Thus, the principal value is:

\[ \cot^{-1}(-\sqrt{3}) + 2 \csc^{-1}(-2) + \cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) = \frac{7\pi}{6} \]

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