Find the solution of `sin^(-1)(x/(1+x))-"sin"^(-1)(x-1)/(x+1)="sin"^(-1)1/(sqrt(1+x))`

Explanation

To solve the equation

\[ \sin^{-1}\left(\frac{x}{1+x}\right) - \sin^{-1}\left(\frac{x-1}{x+1}\right) = \sin^{-1}\left(\frac{1}{\sqrt{1+x}}\right), \]

we will use the property of inverse sine functions.

### Step 1: Rewrite the left-hand side using the property of inverse sine

We know that:

\[ \sin^{-1}(a) - \sin^{-1}(b) = \sin^{-1}\left(a \sqrt{1-b^2} - b \sqrt{1-a^2}\right), \]

where \( a = \frac{x}{1+x} \) and \( b = \frac{x-1}{x+1} \).

### Step 2: Calculate \( 1 - b^2 \) and \( 1 - a^2 \)

First, we need to calculate \( 1 - b^2 \) and \( 1 - a^2 \):

1. **For \( b \)**:
\[ b = \frac{x-1}{x+1} \implies b^2 = \left(\frac{x-1}{x+1}\right)^2 = \frac{(x-1)^2}{(x+1)^2}. \]
Therefore,
\[ 1 - b^2 = 1 - \frac{(x-1)^2}{(x+1)^2} = \frac{(x+1)^2 - (x-1)^2}{(x+1)^2} = \frac{(x^2 + 2x + 1) - (x^2 - 2x + 1)}{(x+1)^2} = \frac{4x}{(x+1)^2}. \]

2. **For \( a \)**:
\[ a = \frac{x}{1+x} \implies a^2 = \left(\frac{x}{1+x}\right)^2 = \frac{x^2}{(1+x)^2}. \]
Therefore,
\[ 1 - a^2 = 1 - \frac{x^2}{(1+x)^2} = \frac{(1+x)^2 - x^2}{(1+x)^2} = \frac{1 + 2x + x^2 - x^2}{(1+x)^2} = \frac{1 + 2x}{(1+x)^2}. \]

### Step 3: Substitute back into the equation

Now substituting \( a \) and \( b \) into the property:

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

### Step 4: Simplify the left-hand side

Substituting the values we calculated:

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

This simplifies to:

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

### Step 5: Set the two sides equal

Now we can set the left-hand side equal to the right-hand side:

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

### Step 6: Cross-multiply and solve for \( x \)

Cross-multiplying gives:

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

### Step 7: Solve the resulting equation

This leads to a polynomial equation in \( x \). Solving this polynomial will yield the values of \( x \).

### Conclusion

The solution of the equation is \( x \geq 0 \).