If `[cot^(-1)x]+[cos^(-1)x]=0` then complete set of values of 'x' is (where [*] denotes the greatest integer function)

Options

A. `(cos 1,1]`
B. `(cot 1, cos 1)`
C. `(cot 1, 1]`
D. None of these

Answer

To solve the equation \([cot^{-1} x] + [cos^{-1} x] = 0\), where \([\cdot]\) denotes the greatest integer function, we will proceed step by step.

### Step 1: Understand the ranges of the functions involved
The function \(cot^{-1} x\) has a range of \((0, \pi)\) and the function \(cos^{-1} x\) has a range of \([0, \pi]\).

**Hint:** Identify the ranges of the inverse trigonometric functions to understand their behavior.

### Step 2: Set up the equation
Given the equation \([cot^{-1} x] + [cos^{-1} x] = 0\), we can denote:
- Let \(y_1 = [cot^{-1} x]\)
- Let \(y_2 = [cos^{-1} x]\)

This means \(y_1 + y_2 = 0\).

**Hint:** Express the greatest integer values in terms of the ranges of the functions.

### Step 3: Analyze the implications of the equation
Since \(y_1\) and \(y_2\) are both integers, for their sum to equal zero, we must have:
- \(y_1 = 0\) and \(y_2 = 0\)

This means:
- \(0 \leq cot^{-1} x < 1\)
- \(0 \leq cos^{-1} x < 1\)

**Hint:** Consider what values of \(x\) would yield \(cot^{-1} x\) and \(cos^{-1} x\) in the specified ranges.

### Step 4: Solve for \(x\) from \(cot^{-1} x\)
From \(0 \leq cot^{-1} x < 1\), we can deduce:
- \(cot^{-1} x = 0\) when \(x \to \infty\)
- \(cot^{-1} x = 1\) when \(x = 1\)

Thus, \(x\) must be in the interval \( (1, \infty) \).

**Hint:** Determine the values of \(x\) that satisfy the conditions derived from \(cot^{-1} x\).

### Step 5: Solve for \(x\) from \(cos^{-1} x\)
From \(0 \leq cos^{-1} x < 1\), we know:
- \(cos^{-1} x = 0\) when \(x = 1\)
- \(cos^{-1} x = 1\) when \(x = cos(1)\)

Thus, \(x\) must be in the interval \( (cos(1), 1] \).

**Hint:** Analyze the implications of the range for \(cos^{-1} x\) to find valid \(x\) values.

### Step 6: Combine the results
Now we need to find the intersection of the two intervals:
- From \(cot^{-1} x\): \(x \in (1, \infty)\)
- From \(cos^{-1} x\): \(x \in (cos(1), 1]\)

The only value that satisfies both conditions is \(x = 1\).

**Hint:** Look for common values in the derived intervals.

### Final Answer
Thus, the complete set of values of \(x\) that satisfies the equation \([cot^{-1} x] + [cos^{-1} x] = 0\) is:
\[ x = 1 \]