The sum of solutions of the equation `2 sin^(-1) sqrt(x^(2)+x+1)+cos^(-1) sqrt(x^(2)+x)=(3pi)/(2)` is :

Options

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

Answer

To solve the equation \( 2 \sin^{-1} \sqrt{x^2 + x + 1} + \cos^{-1} \sqrt{x^2 + x} = \frac{3\pi}{2} \), we will follow these steps:

### Step 1: Rewrite the equation
We can rewrite the equation as:
\[ 2 \sin^{-1} \sqrt{x^2 + x + 1} = \frac{3\pi}{2} - \cos^{-1} \sqrt{x^2 + x} \]

### Step 2: Use the identity for inverse trigonometric functions
Using the identity \( \cos^{-1}(y) + \sin^{-1}(y) = \frac{\pi}{2} \), we can express \( \cos^{-1} \sqrt{x^2 + x} \) in terms of \( \sin^{-1} \):
\[ \cos^{-1} \sqrt{x^2 + x} = \frac{\pi}{2} - \sin^{-1} \sqrt{x^2 + x} \]

### Step 3: Substitute back into the equation
Substituting this into our equation gives:
\[ 2 \sin^{-1} \sqrt{x^2 + x + 1} = \frac{3\pi}{2} - \left(\frac{\pi}{2} - \sin^{-1} \sqrt{x^2 + x}\right) \]
This simplifies to:
\[ 2 \sin^{-1} \sqrt{x^2 + x + 1} = \pi + \sin^{-1} \sqrt{x^2 + x} \]

### Step 4: Isolate the inverse sine terms
Rearranging the equation, we have:
\[ 2 \sin^{-1} \sqrt{x^2 + x + 1} - \sin^{-1} \sqrt{x^2 + x} = \pi \]

### Step 5: Let \( y = \sin^{-1} \sqrt{x^2 + x} \)
Let \( y = \sin^{-1} \sqrt{x^2 + x} \). Then we can express \( \sqrt{x^2 + x + 1} \) in terms of \( y \):
\[ \sqrt{x^2 + x} = \sin y \implies x^2 + x = \sin^2 y \implies x^2 + x - \sin^2 y = 0 \]
Using the quadratic formula, we find:
\[ x = \frac{-1 \pm \sqrt{1 + 4\sin^2 y}}{2} \]

### Step 6: Substitute back into the equation
Now substituting back into the equation gives:
\[ 2 \sin^{-1} \sqrt{\frac{1 + 4\sin^2 y}{4}} - y = \pi \]
This can be simplified further, but we need to find the values of \( y \) that satisfy this equation.

### Step 7: Solve for \( y \)
This equation can be solved numerically or graphically to find the values of \( y \) that satisfy it.

### Step 8: Find the corresponding \( x \) values
Once we have the values of \( y \), we can substitute back to find the corresponding \( x \) values using:
\[ x = \frac{-1 \pm \sqrt{1 + 4\sin^2 y}}{2} \]

### Step 9: Sum the solutions
Finally, we sum the solutions for \( x \) obtained from the values of \( y \).