Solve the equation for x :
`"sin"^(-1)(5)/(x)+"sin"^(-1)(12)/(x)=(pi)/(2), x ne 0`.

Answer

To solve the equation

\[ \sin^{-1}\left(\frac{5}{x}\right) + \sin^{-1}\left(\frac{12}{x}\right) = \frac{\pi}{2}, \quad x \neq 0, \]

we can follow these steps:

### Step 1: Set up the equation
Let
\[ \beta = \sin^{-1}\left(\frac{12}{x}\right). \]
Then, we can express the first term as:
\[ \sin^{-1}\left(\frac{5}{x}\right) + \beta = \frac{\pi}{2}. \]

### Step 2: Rearranging the equation
From the equation, we can rearrange it to find:
\[ \sin^{-1}\left(\frac{5}{x}\right) = \frac{\pi}{2} - \beta. \]

### Step 3: Use the sine of complementary angles
Using the identity \(\sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta)\), we have:
\[ \frac{5}{x} = \cos(\beta). \]

### Step 4: Express \(\cos(\beta)\)
Since \(\beta = \sin^{-1}\left(\frac{12}{x}\right)\), we can find \(\cos(\beta)\) using the Pythagorean identity:
\[ \cos(\beta) = \sqrt{1 - \sin^2(\beta)} = \sqrt{1 - \left(\frac{12}{x}\right)^2} = \sqrt{1 - \frac{144}{x^2}}. \]

### Step 5: Substitute \(\cos(\beta)\) back
Substituting back into the equation gives:
\[ \frac{5}{x} = \sqrt{1 - \frac{144}{x^2}}. \]

### Step 6: Square both sides
Squaring both sides to eliminate the square root yields:
\[ \left(\frac{5}{x}\right)^2 = 1 - \frac{144}{x^2}. \]
This simplifies to:
\[ \frac{25}{x^2} = 1 - \frac{144}{x^2}. \]

### Step 7: Combine terms
Multiplying through by \(x^2\) to eliminate the denominators gives:
\[ 25 = x^2 - 144. \]

### Step 8: Rearranging the equation
Rearranging this gives:
\[ x^2 = 25 + 144 = 169. \]

### Step 9: Solve for \(x\)
Taking the square root of both sides yields:
\[ x = \pm 13. \]

### Step 10: Consider the restriction
Since \(x \neq 0\), both \(x = 13\) and \(x = -13\) are valid solutions.

Thus, the final solutions are:
\[ x = 13 \quad \text{and} \quad x = -13. \]