The value of `cot^(-1)(-sqrt3)+cosec^(-1)(2)+tan^(-1)(sqrt3)` is

Options

A. `pi/6`
B. `pi/3`
C. `(5pi)/6`
D. `(4pi)/3`

Answer

To solve the expression \( \cot^{-1}(-\sqrt{3}) + \csc^{-1}(2) + \tan^{-1}(\sqrt{3}) \), we will evaluate each term step by step.

### Step 1: Evaluate \( \cot^{-1}(-\sqrt{3}) \)
Using the property of inverse cotangent:
\[ \cot^{-1}(-x) = \pi - \cot^{-1}(x) \]
we can rewrite:
\[ \cot^{-1}(-\sqrt{3}) = \pi - \cot^{-1}(\sqrt{3}) \]
Next, we know that:
\[ \cot^{-1}(\sqrt{3}) = \frac{\pi}{6} \]
Thus:
\[ \cot^{-1}(-\sqrt{3}) = \pi - \frac{\pi}{6} = \frac{6\pi}{6} - \frac{\pi}{6} = \frac{5\pi}{6} \]

### Step 2: Evaluate \( \csc^{-1}(2) \)
The cosecant function is the reciprocal of sine. Therefore:
\[ \csc^{-1}(2) = \sin^{-1}\left(\frac{1}{2}\right) \]
We know that:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
Thus:
\[ \csc^{-1}(2) = \frac{\pi}{6} \]

### Step 3: Evaluate \( \tan^{-1}(\sqrt{3}) \)
We know that:
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]
Thus:
\[ \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \]

### Step 4: Combine all the values
Now we can combine the results from the previous steps:
\[ \cot^{-1}(-\sqrt{3}) + \csc^{-1}(2) + \tan^{-1}(\sqrt{3}) = \frac{5\pi}{6} + \frac{\pi}{6} + \frac{\pi}{3} \]
To add these fractions, we need a common denominator. The common denominator is 6:
\[ \frac{5\pi}{6} + \frac{\pi}{6} + \frac{2\pi}{6} = \frac{5\pi + \pi + 2\pi}{6} = \frac{8\pi}{6} = \frac{4\pi}{3} \]

### Final Answer
The value of \( \cot^{-1}(-\sqrt{3}) + \csc^{-1}(2) + \tan^{-1}(\sqrt{3}) \) is:
\[ \frac{4\pi}{3} \]