What is the value of `"tan"^(-1) sqrt3-cot^(-1)(-sqrt3).`

Options

A. `pi/2`
B. `- pi/2`
C. 0
D. 1

Answer

To solve the expression \( \tan^{-1}(\sqrt{3}) - \cot^{-1}(-\sqrt{3}) \), we can follow these steps:

### Step 1: Rewrite the cotangent inverse term
We know that:
\[ \cot^{-1}(-x) = \pi - \cot^{-1}(x) \]
Applying this to our expression, we get:
\[ \cot^{-1}(-\sqrt{3}) = \pi - \cot^{-1}(\sqrt{3}) \]
Thus, we can rewrite the expression as:
\[ \tan^{-1}(\sqrt{3}) - (\pi - \cot^{-1}(\sqrt{3})) \]

### Step 2: Simplify the expression
Substituting the rewritten cotangent inverse back into the expression gives:
\[ \tan^{-1}(\sqrt{3}) - \pi + \cot^{-1}(\sqrt{3}) \]
Now we can combine the terms:
\[ \tan^{-1}(\sqrt{3}) + \cot^{-1}(\sqrt{3}) - \pi \]

### Step 3: Apply the identity
We know from trigonometric identities that:
\[ \tan^{-1}(x) + \cot^{-1}(x) = \frac{\pi}{2} \]
So, applying this identity for \( x = \sqrt{3} \):
\[ \tan^{-1}(\sqrt{3}) + \cot^{-1}(\sqrt{3}) = \frac{\pi}{2} \]

### Step 4: Substitute back into the expression
Now substituting this back into our expression:
\[ \frac{\pi}{2} - \pi \]

### Step 5: Simplify the final expression
This simplifies to:
\[ \frac{\pi}{2} - \frac{2\pi}{2} = -\frac{\pi}{2} \]

### Final Answer
Thus, the value of \( \tan^{-1}(\sqrt{3}) - \cot^{-1}(-\sqrt{3}) \) is:
\[ -\frac{\pi}{2} \]