The number of real solution of `cot^(-1)sqrt(x(x+4))+cos^(-1)sqrt(x^(2)+4x+1)=(pi)/(2)` is equal to

Options

A. 0
B. 1
C. 2
D. Infinite

Answer

To solve the equation

\[ \cot^{-1}(\sqrt{x(x+4)}) + \cos^{-1}(\sqrt{x^2 + 4x + 1}) = \frac{\pi}{2}, \]

we will follow these steps:

### Step 1: Rewrite the equation in terms of a single variable
Let

\[ \theta = \cot^{-1}(\sqrt{x(x+4)}). \]

Then, we can express the equation as:

\[ \theta + \cos^{-1}(\sqrt{x^2 + 4x + 1}) = \frac{\pi}{2}. \]

### Step 2: Use the identity for inverse trigonometric functions
Using the identity

\[ \sin^{-1}(y) + \cos^{-1}(y) = \frac{\pi}{2}, \]

we can rewrite our equation as:

\[ \sin^{-1}\left(\frac{1}{\sqrt{x^2 + 4x + 1}}\right) + \cos^{-1}\left(\sqrt{x^2 + 4x + 1}\right) = \frac{\pi}{2}. \]

This implies that

\[ \frac{1}{\sqrt{x^2 + 4x + 1}} = \sqrt{x^2 + 4x + 1}. \]

### Step 3: Square both sides to eliminate the square root
Squaring both sides gives:

\[ \frac{1}{x^2 + 4x + 1} = x^2 + 4x + 1. \]

### Step 4: Rearrange the equation
Multiplying both sides by \(x^2 + 4x + 1\) (assuming it's not zero) results in:

\[ 1 = (x^2 + 4x + 1)^2. \]

### Step 5: Expand and simplify
Expanding the right side:

\[ 1 = x^4 + 8x^3 + 18x^2 + 8x + 1. \]

Subtracting 1 from both sides leads to:

\[ 0 = x^4 + 8x^3 + 18x^2 + 8x. \]

### Step 6: Factor out common terms
Factoring out \(x\):

\[ x(x^3 + 8x^2 + 18x + 8) = 0. \]

This gives us one solution:

\[ x = 0. \]

### Step 7: Solve the cubic equation
Now we need to find the roots of the cubic equation:

\[ x^3 + 8x^2 + 18x + 8 = 0. \]

Using the Rational Root Theorem, we can test possible rational roots. Testing \(x = -4\):

\[ (-4)^3 + 8(-4)^2 + 18(-4) + 8 = -64 + 128 - 72 + 8 = 0. \]

Thus, \(x = -4\) is a root.

### Step 8: Factor the cubic polynomial
Now we can factor the cubic polynomial as:

\[ (x + 4)(x^2 + 4x + 2) = 0. \]

### Step 9: Find the roots of the quadratic
To find the roots of \(x^2 + 4x + 2 = 0\), we use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-4 \pm \sqrt{16 - 8}}{2} = \frac{-4 \pm 2\sqrt{2}}{2} = -2 \pm \sqrt{2}. \]

### Step 10: List all solutions
The solutions to the original equation are:

1. \(x = 0\)
2. \(x = -4\)
3. \(x = -2 + \sqrt{2}\)
4. \(x = -2 - \sqrt{2}\)

### Conclusion
Thus, the total number of real solutions is **4**.