Find `x` satisfying `[tan^(-1)x]+[cos^(-1)x]=2,` where `[]` represents the greatest integer function.

Explanation

To solve the equation \([tan^{-1}x] + [cos^{-1}x] = 2\), where \([]\) denotes the greatest integer function, we need to analyze the ranges of the inverse trigonometric functions involved.

### Step 1: Understanding the Ranges
1. The range of \(tan^{-1}x\) is \((- \frac{\pi}{2}, \frac{\pi}{2})\).
2. The range of \(cos^{-1}x\) is \([0, \pi]\).

### Step 2: Analyzing the Greatest Integer Function
The greatest integer function \([y]\) gives the largest integer less than or equal to \(y\). Therefore, we can deduce the possible values of \([tan^{-1}x]\) and \([cos^{-1}x]\).
...