The number of real roots of the equation `tan^(-1)sqrt(x(x+1))+sin^(-1)sqrt(x^(2)+x+1)=(pi)/(2)` is :

Answer

To solve the equation

\[ \tan^{-1}(\sqrt{x(x+1)}) + \sin^{-1}(\sqrt{x^2 + x + 1}) = \frac{\pi}{2}, \]

we can follow these steps:

### Step 1: Rewrite the Equation
Let \( y = \tan^{-1}(\sqrt{x(x+1)}) \). Then, we can rewrite the equation as:

\[ y + \sin^{-1}(\sqrt{x^2 + x + 1}) = \frac{\pi}{2}. \]

### Step 2: Use the Identity
From the identity \( \sin^{-1}(a) + \cos^{-1}(a) = \frac{\pi}{2} \), we can express \( \sin^{-1}(\sqrt{x^2 + x + 1}) \) as:

\[ \sin^{-1}(\sqrt{x^2 + x + 1}) = \frac{\pi}{2} - y. \]

This implies:

\[ y = \cos^{-1}(\sqrt{x^2 + x + 1}). \]

### Step 3: Set the Arguments Equal
Since both expressions for \( y \) are equal, we can set the arguments equal:

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

### Step 4: Square Both Sides
Squaring both sides gives us:

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

### Step 5: Simplify the Equation
Rearranging the equation, we have:

\[ x^2 + x = x^2 + x + 1 \implies 0 = 1. \]

This is a contradiction, which indicates that we need to check the conditions under which the original equation holds.

### Step 6: Analyze the Function
We need to analyze the function:

\[ f(x) = \tan^{-1}(\sqrt{x(x+1)}) + \sin^{-1}(\sqrt{x^2 + x + 1}). \]

### Step 7: Determine the Range of \( f(x) \)
1. **Domain**: The expressions inside the inverse functions must be defined.
- For \( \tan^{-1}(\sqrt{x(x+1)}) \), \( x(x+1) \geq 0 \) implies \( x \geq 0 \) or \( x \leq -1 \).
- For \( \sin^{-1}(\sqrt{x^2 + x + 1}) \), since \( \sqrt{x^2 + x + 1} \) is always positive and less than or equal to 1, it is valid for all \( x \).

2. **Behavior**: As \( x \to \infty \), both \( \tan^{-1} \) and \( \sin^{-1} \) approach their limits, and we can analyze the behavior at critical points like \( x = 0 \) and \( x = -1 \).

### Step 8: Find the Roots
From our analysis, we find that the only values that satisfy the equation are \( x = 0 \) and \( x = -1 \).

### Conclusion
Thus, the number of real roots of the given equation is **2** (the roots are \( x = 0 \) and \( x = -1 \)).