Write the principal value of `tan^(-1) (sqrt(3)) +"cosec"^(-1) (-2)`

Answer

To find the principal value of \( \tan^{-1}(\sqrt{3}) + \csc^{-1}(-2) \), we can follow these steps:

### Step 1: Rewrite the expression
We can rewrite \( \csc^{-1}(-2) \) using the identity for the inverse cosecant function:
\[ \csc^{-1}(-x) = -\csc^{-1}(x) \]
Thus, we have:
\[ \csc^{-1}(-2) = -\csc^{-1}(2) \]
So, the expression becomes:
\[ \tan^{-1}(\sqrt{3}) - \csc^{-1}(2) \]

### Step 2: Find \( \tan^{-1}(\sqrt{3}) \)
We know that:
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]
Therefore:
\[ \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \]

### Step 3: Find \( \csc^{-1}(2) \)
The cosecant function is the reciprocal of the sine function. Therefore:
\[ \csc^{-1}(2) = \theta \quad \text{where} \quad \csc(\theta) = 2 \implies \sin(\theta) = \frac{1}{2} \]
The angle \( \theta \) that satisfies this in the range of the cosecant function is:
\[ \theta = \frac{\pi}{6} \]
Thus:
\[ \csc^{-1}(2) = \frac{\pi}{6} \]

### Step 4: Substitute the values back into the expression
Now, substituting back into our expression:
\[ \tan^{-1}(\sqrt{3}) - \csc^{-1}(2) = \frac{\pi}{3} - \frac{\pi}{6} \]

### Step 5: Simplify the expression
To simplify \( \frac{\pi}{3} - \frac{\pi}{6} \), we need a common denominator:
\[ \frac{\pi}{3} = \frac{2\pi}{6} \]
Thus:
\[ \frac{2\pi}{6} - \frac{\pi}{6} = \frac{2\pi - \pi}{6} = \frac{\pi}{6} \]

### Final Answer
The principal value of \( \tan^{-1}(\sqrt{3}) + \csc^{-1}(-2) \) is:
\[ \frac{\pi}{6} \]