The solution set of inequality
`(tan^(-1)x)(cot^(-1)x)-(tan^(-1)x)(1+(pi)/(2))-2cot^(-1)x+2(1+(pi)/(2))gtlim_(yrarr-oo)[sec^(-1)y-(pi)/(2)]` is (where [ . ]denotes the G.I.F.)

Options

A. `(tan1, tan2)`
B. `(-cot1, cot2)`
C. `(-tan1, tan2)`
D. `(-tan1, oo)`

Explanation

To solve the inequality

\[ (\tan^{-1} x)(\cot^{-1} x) - (\tan^{-1} x)(1 + \frac{\pi}{2}) - 2\cot^{-1} x + 2(1 + \frac{\pi}{2}) > \lim_{y \to -\infty} \left[ \sec^{-1} y - \frac{\pi}{2} \right] \]

we will break it down step by step.

### Step 1: Evaluate the Right-Hand Side
First, we need to evaluate the limit on the right-hand side:

\[ \lim_{y \to -\infty} \left[ \sec^{-1} y - \frac{\pi}{2} \right] \]

As \( y \to -\infty \), \( \sec^{-1} y \) approaches \( \frac{\pi}{2} \). Therefore,

\[ \sec^{-1}(-\infty) = \frac{\pi}{2} \]

Thus, the limit becomes:

\[ \lim_{y \to -\infty} \left[ \sec^{-1} y - \frac{\pi}{2} \right] = \frac{\pi}{2} - \frac{\pi}{2} = 0 \]

### Step 2: Rewrite the Inequality
Now we can rewrite the inequality:

\[ (\tan^{-1} x)(\cot^{-1} x) - (\tan^{-1} x)(1 + \frac{\pi}{2}) - 2\cot^{-1} x + 2(1 + \frac{\pi}{2}) > 0 \]

### Step 3: Factor the Expression
Let's factor out common terms. We can take \( \cot^{-1} x \) common from the first and third terms, and \( 1 + \frac{\pi}{2} \) from the second and fourth terms:

\[ \cot^{-1} x (\tan^{-1} x - 2) + 2(1 + \frac{\pi}{2}) - \tan^{-1} x(1 + \frac{\pi}{2}) > 0 \]

### Step 4: Further Simplify
Now, we can rearrange the terms:

\[ (\tan^{-1} x - 2)(\cot^{-1} x - (1 + \frac{\pi}{2})) > 0 \]

### Step 5: Identify Critical Points
The critical points occur when:

1. \( \tan^{-1} x - 2 = 0 \) which gives \( \tan^{-1} x = 2 \) or \( x = \tan(2) \)
2. \( \cot^{-1} x - (1 + \frac{\pi}{2}) = 0 \) gives \( \cot^{-1} x = 1 + \frac{\pi}{2} \) or \( x = \cot(1 + \frac{\pi}{2}) \)

### Step 6: Analyze the Intervals
We need to analyze the sign of the expression in the intervals defined by the critical points:

1. For \( x < \tan(2) \)
2. For \( \tan(2) < x < \cot(1 + \frac{\pi}{2}) \)
3. For \( x > \cot(1 + \frac{\pi}{2}) \)

### Step 7: Determine the Solution Set
After testing the intervals, we find that the expression is positive in the intervals where:

\[ \tan^{-1} x \in (-1, 2) \]

### Step 8: Convert Back to x
Finally, we convert the intervals back to \( x \):

1. \( \tan^{-1}(-1) < x < \tan^{-1}(2) \)
2. This gives us \( -\tan(1) < x < \tan(2) \)

Thus, the solution set is:

\[ x \in (-\tan(1), \tan(2)) \]