The sum of absolute value of all possible values of `x` for which `cos tan^(-1) sin cot^(-1)x=sqrt(226/227)`.

Explanation

To solve the equation \( \cos(\tan^{-1}(\sin(\cot^{-1}(x)))) = \sqrt{\frac{226}{227}} \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \cos(\tan^{-1}(\sin(\cot^{-1}(x)))) \]
We know that \( \cot^{-1}(x) \) can be expressed in terms of sine. Specifically, we have:
\[ \sin(\cot^{-1}(x)) = \frac{x}{\sqrt{1+x^2}} \]
Thus, we can rewrite our expression as:
\[ \cos(\tan^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right)) \]

### Step 2: Use the identity for cosine of arctangent
Using the identity \( \cos(\tan^{-1}(y)) = \frac{1}{\sqrt{1+y^2}} \), we can substitute \( y = \frac{x}{\sqrt{1+x^2}} \):
\[ \cos(\tan^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right)) = \frac{1}{\sqrt{1+\left(\frac{x}{\sqrt{1+x^2}}\right)^2}} \]
Calculating \( \left(\frac{x}{\sqrt{1+x^2}}\right)^2 \):
\[ \left(\frac{x}{\sqrt{1+x^2}}\right)^2 = \frac{x^2}{1+x^2} \]
So we have:
\[ 1+\left(\frac{x}{\sqrt{1+x^2}}\right)^2 = 1 + \frac{x^2}{1+x^2} = \frac{1+x^2+x^2}{1+x^2} = \frac{1+2x^2}{1+x^2} \]
Thus:
\[ \cos(\tan^{-1}(\sin(\cot^{-1}(x)))) = \frac{1}{\sqrt{\frac{1+2x^2}{1+x^2}}} = \frac{\sqrt{1+x^2}}{\sqrt{1+2x^2}} \]

### Step 3: Set the equation
Now we set the expression equal to \( \sqrt{\frac{226}{227}} \):
\[ \frac{\sqrt{1+x^2}}{\sqrt{1+2x^2}} = \sqrt{\frac{226}{227}} \]

### Step 4: Square both sides
Squaring both sides gives:
\[ \frac{1+x^2}{1+2x^2} = \frac{226}{227} \]

### Step 5: Cross-multiply
Cross-multiplying yields:
\[ 227(1+x^2) = 226(1+2x^2) \]
Expanding both sides:
\[ 227 + 227x^2 = 226 + 452x^2 \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ 227 - 226 = 452x^2 - 227x^2 \]
This simplifies to:
\[ 1 = 225x^2 \]
Thus:
\[ x^2 = \frac{1}{225} \]

### Step 7: Solve for x
Taking the square root gives:
\[ x = \pm \frac{1}{15} \]

### Step 8: Find the sum of absolute values
The absolute values of \( x \) are:
\[ |x| = \frac{1}{15} \]
Thus, the sum of the absolute values of all possible values of \( x \) is:
\[ \frac{1}{15} + \frac{1}{15} = \frac{2}{15} \]

### Final Answer
The final answer is:
\[ \text{Sum of absolute values} = \frac{2}{15} \]