If `a^(2) + b^(2) = c^(2), c != 0`, then find the non-zero solution of the equation:
`sin^(-1).(ax)/(c) + sin^(-1).(bx)/(c) = sin^(-1) x`

Explanation

To solve the equation \( \sin^{-1} \left( \frac{ax}{c} \right) + \sin^{-1} \left( \frac{bx}{c} \right) = \sin^{-1}(x) \), given that \( a^2 + b^2 = c^2 \) and \( c \neq 0 \), we can follow these steps:

### Step 1: Use the formula for the sum of inverse sine functions
We know that:
\[ \sin^{-1}(u) + \sin^{-1}(v) = \sin^{-1}(u \sqrt{1 - v^2} + v \sqrt{1 - u^2}) \]
Applying this to our equation, we set \( u = \frac{ax}{c} \) and \( v = \frac{bx}{c} \):
\[ \sin^{-1} \left( \frac{ax}{c} \sqrt{1 - \left( \frac{bx}{c} \right)^2} + \frac{bx}{c} \sqrt{1 - \left( \frac{ax}{c} \right)^2} \right) = \sin^{-1}(x) \]

### Step 2: Simplify the left-hand side
The left-hand side becomes:
\[ \frac{ax}{c} \sqrt{1 - \frac{b^2x^2}{c^2}} + \frac{bx}{c} \sqrt{1 - \frac{a^2x^2}{c^2}} \]
This simplifies to:
\[ \frac{ax}{c} \sqrt{\frac{c^2 - b^2x^2}{c^2}} + \frac{bx}{c} \sqrt{\frac{c^2 - a^2x^2}{c^2}} \]
\[ = \frac{ax \sqrt{c^2 - b^2x^2}}{c} + \frac{bx \sqrt{c^2 - a^2x^2}}{c} \]

### Step 3: Set the equation
Now we have:
\[ \frac{ax \sqrt{c^2 - b^2x^2} + bx \sqrt{c^2 - a^2x^2}}{c} = x \]

### Step 4: Multiply both sides by \( c \)
Multiplying both sides by \( c \):
\[ ax \sqrt{c^2 - b^2x^2} + bx \sqrt{c^2 - a^2x^2} = cx \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ ax \sqrt{c^2 - b^2x^2} + bx \sqrt{c^2 - a^2x^2} - cx = 0 \]

### Step 6: Factor out \( x \)
Factoring out \( x \):
\[ x \left( a \sqrt{c^2 - b^2x^2} + b \sqrt{c^2 - a^2x^2} - c \right) = 0 \]
Since \( x \neq 0 \), we can set:
\[ a \sqrt{c^2 - b^2x^2} + b \sqrt{c^2 - a^2x^2} - c = 0 \]

### Step 7: Square both sides
Squaring both sides:
\[ \left( a \sqrt{c^2 - b^2x^2} + b \sqrt{c^2 - a^2x^2} \right)^2 = c^2 \]
Expanding gives:
\[ a^2(c^2 - b^2x^2) + b^2(c^2 - a^2x^2) + 2ab \sqrt{(c^2 - b^2x^2)(c^2 - a^2x^2)} = c^2 \]

### Step 8: Substitute \( c^2 = a^2 + b^2 \)
Using \( c^2 = a^2 + b^2 \):
\[ a^2(c^2 - b^2x^2) + b^2(c^2 - a^2x^2) = c^2 - 2ab \sqrt{(c^2 - b^2x^2)(c^2 - a^2x^2)} \]

### Step 9: Solve for \( x^2 \)
After simplification, we find:
\[ x^2 = 1 \]
Thus, the solutions for \( x \) are:
\[ x = \pm 1 \]

### Final Answer
The non-zero solutions of the equation are:
\[ x = 1 \quad \text{and} \quad x = -1 \]