If `cos(tan^(-1)x+cot^(-1)sqrt(3))=0` , find the value of `xdot`

Explanation

To solve the equation \( \cos(\tan^{-1} x + \cot^{-1} \sqrt{3}) = 0 \), we will follow these steps:

### Step 1: Set up the equation
We start with the given equation:
\[ \cos(\tan^{-1} x + \cot^{-1} \sqrt{3}) = 0 \]

### Step 2: Use the property of cosine
We know that \( \cos(\theta) = 0 \) when \( \theta = \frac{\pi}{2} + n\pi \) for any integer \( n \). For our case, we can take:
\[ \tan^{-1} x + \cot^{-1} \sqrt{3} = \frac{\pi}{2} \]

### Step 3: Use the identity
We know from trigonometric identities that:
\[ \tan^{-1} x + \cot^{-1} y = \frac{\pi}{2} \quad \text{if } y = \frac{1}{x} \]
In our case, we have \( y = \sqrt{3} \). Thus, we can write:
\[ \tan^{-1} x + \cot^{-1} \sqrt{3} = \frac{\pi}{2} \]

### Step 4: Relate \( x \) to \( \sqrt{3} \)
From the identity, we can conclude that:
\[ \sqrt{3} = \frac{1}{x} \]
This implies:
\[ x = \frac{1}{\sqrt{3}} \]

### Step 5: Rationalize the denominator
To express \( x \) in a more standard form, we can rationalize the denominator:
\[ x = \frac{\sqrt{3}}{3} \]

### Final Answer
Thus, the value of \( x \) is:
\[ x = \frac{\sqrt{3}}{3} \]

---