If `((x+1)^(2))/(x^(3)+x)=(A)/(x)+(Bx+C)/(x^(2)+1)`, then
`cosec^(-1)((1)/(A))+cot^(-1).(1)/(B)+sec^(-1)C` = _________

Options

A. `(pi)/(6)`
B. `(5pi)/(6)`
C. `(pi)/(2)`
D. 0

Answer

To solve the equation

\[ \frac{(x+1)^2}{x^3+x} = \frac{A}{x} + \frac{Bx+C}{x^2+1}, \]

we will follow these steps:

### Step 1: Simplify the Left Side
First, simplify the left side of the equation:

\[ x^3 + x = x(x^2 + 1). \]

Thus, we can rewrite the left-hand side as:

\[ \frac{(x+1)^2}{x(x^2+1)}. \]

### Step 2: Rewrite the Right Side
Now, we rewrite the right side of the equation with a common denominator:

\[ \frac{A}{x} + \frac{Bx+C}{x^2+1} = \frac{A(x^2+1) + (Bx+C)x}{x(x^2+1)}. \]

### Step 3: Combine the Right Side
Combine the terms in the numerator of the right side:

\[ A(x^2 + 1) + (Bx + C)x = Ax^2 + A + Bx^2 + Cx = (A + B)x^2 + Cx + A. \]

### Step 4: Set the Numerators Equal
Now, we have:

\[ \frac{(x+1)^2}{x(x^2+1)} = \frac{(A + B)x^2 + Cx + A}{x(x^2+1)}. \]

Since the denominators are the same, we can equate the numerators:

\[ (x+1)^2 = (A + B)x^2 + Cx + A. \]

### Step 5: Expand the Left Side
Expanding the left side gives:

\[ x^2 + 2x + 1. \]

### Step 6: Compare Coefficients
Now we equate coefficients from both sides:

1. Coefficient of \(x^2\): \(1 = A + B\)
2. Coefficient of \(x\): \(2 = C\)
3. Constant term: \(1 = A\)

### Step 7: Solve the System of Equations
From the equations:

1. \(A = 1\)
2. \(C = 2\)
3. Substitute \(A\) into the first equation: \(1 = 1 + B \Rightarrow B = 0\)

Thus, we have:

- \(A = 1\)
- \(B = 0\)
- \(C = 2\)

### Step 8: Substitute Values into the Expression
Now we substitute \(A\), \(B\), and \(C\) into the expression:

\[ \csc^{-1}\left(\frac{1}{A}\right) + \cot^{-1}\left(\frac{1}{B}\right) + \sec^{-1}(C). \]

This becomes:

\[ \csc^{-1}(1) + \cot^{-1}(\text{undefined}) + \sec^{-1}(2). \]

### Step 9: Evaluate Each Term
1. \(\csc^{-1}(1) = \frac{\pi}{2}\)
2. \(\cot^{-1}(0)\) is undefined.
3. \(\sec^{-1}(2) = \frac{\pi}{3}\).

### Step 10: Combine the Results
Since \(\cot^{-1}(0)\) is undefined, we consider the other two terms:

\[ \frac{\pi}{2} + \frac{\pi}{3} = \frac{3\pi}{6} + \frac{2\pi}{6} = \frac{5\pi}{6}. \]

### Final Answer
Thus, the final answer is:

\[ \frac{5\pi}{6}. \]