Solve the following equations: `sin[2cos^(-1)"{"cot"("2tan^(-1)x"}]"=0`

Options

A. `pm1`
B. `1pm sqrt(2)`
C. `-1pm sqrt(2)`
D. `pm sqrt(2)`

Explanation

To solve the equation \( \sin\left(2 \cos^{-1}\left(\cot\left(2 \tan^{-1}(x)\right)\right)\right) = 0 \), we will follow these steps:

### Step 1: Set the equation for sine
Since \( \sin(y) = 0 \) when \( y = n\pi \) for any integer \( n \), we can set:
\[ 2 \cos^{-1}\left(\cot\left(2 \tan^{-1}(x)\right)\right) = n\pi \]

### Step 2: Divide by 2
Dividing both sides by 2 gives:
\[ \cos^{-1}\left(\cot\left(2 \tan^{-1}(x)\right)\right) = \frac{n\pi}{2} \]

### Step 3: Determine valid values for \( n \)
The range of \( \cos^{-1}(x) \) is \( [0, \pi] \). Therefore, \( \frac{n\pi}{2} \) must also lie within this range. This gives us the possible integer values for \( n \):
- \( n = 0 \) gives \( 0 \)
- \( n = 1 \) gives \( \frac{\pi}{2} \)
- \( n = 2 \) gives \( \pi \)

Thus, \( n \) can be \( 0, 1, \) or \( 2 \).

### Step 4: Case 1: \( n = 0 \)
For \( n = 0 \):
\[ \cos^{-1}\left(\cot\left(2 \tan^{-1}(x)\right)\right) = 0 \]
This implies:
\[ \cot\left(2 \tan^{-1}(x)\right) = \cos(0) = 1 \]
Thus:
\[ 2 \tan^{-1}(x) = \tan^{-1}(1) \implies 2 \tan^{-1}(x) = \frac{\pi}{4} \]
This leads to:
\[ \tan^{-1}(x) = \frac{\pi}{8} \implies x = \tan\left(\frac{\pi}{8}\right) \]

### Step 5: Case 2: \( n = 1 \)
For \( n = 1 \):
\[ \cos^{-1}\left(\cot\left(2 \tan^{-1}(x)\right)\right) = \frac{\pi}{2} \]
This implies:
\[ \cot\left(2 \tan^{-1}(x)\right) = \cos\left(\frac{\pi}{2}\right) = 0 \]
Thus:
\[ 2 \tan^{-1}(x) = \frac{\pi}{2} \implies \tan^{-1}(x) = \frac{\pi}{4} \implies x = 1 \]

### Step 6: Case 3: \( n = 2 \)
For \( n = 2 \):
\[ \cos^{-1}\left(\cot\left(2 \tan^{-1}(x)\right)\right) = \pi \]
This implies:
\[ \cot\left(2 \tan^{-1}(x)\right) = \cos(\pi) = -1 \]
Thus:
\[ 2 \tan^{-1}(x) = \tan^{-1}(-1) \implies 2 \tan^{-1}(x) = -\frac{\pi}{4} \]
This leads to:
\[ \tan^{-1}(x) = -\frac{\pi}{8} \implies x = \tan\left(-\frac{\pi}{8}\right) = -\tan\left(\frac{\pi}{8}\right) \]

### Final Solutions
The solutions for \( x \) from the three cases are:
1. \( x = \tan\left(\frac{\pi}{8}\right) \)
2. \( x = 1 \)
3. \( x = -\tan\left(\frac{\pi}{8}\right) \)

### Summary of Solutions
Thus, the final solutions are:
\[ x = \tan\left(\frac{\pi}{8}\right), \quad x = 1, \quad x = -\tan\left(\frac{\pi}{8}\right) \]