Solve `sin^(-1) (5/x) + sin^(-1)(12/x) = pi/2`

Answer

To solve the equation \( \sin^{-1} \left( \frac{5}{x} \right) + \sin^{-1} \left( \frac{12}{x} \right) = \frac{\pi}{2} \), we can follow these steps:

### Step 1: Use the identity for inverse sine
We know that:
\[ \sin^{-1}(a) + \sin^{-1}(b) = \frac{\pi}{2} \implies a^2 + b^2 = 1 \]
In our case, let \( a = \frac{5}{x} \) and \( b = \frac{12}{x} \). Therefore, we can write:
\[ \left( \frac{5}{x} \right)^2 + \left( \frac{12}{x} \right)^2 = 1 \]

### Step 2: Simplify the equation
Squaring the terms gives:
\[ \frac{25}{x^2} + \frac{144}{x^2} = 1 \]
Combining the fractions:
\[ \frac{25 + 144}{x^2} = 1 \]
This simplifies to:
\[ \frac{169}{x^2} = 1 \]

### Step 3: Cross-multiply to solve for \( x^2 \)
Cross-multiplying gives:
\[ 169 = x^2 \]

### Step 4: Take the square root
Taking the square root of both sides yields:
\[ x = \pm 13 \]

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