If `"sin"^(-1)(5)/(x) +"sin"^(-1)(12)/(x)=(pi)/(2)`, then what is the value of x?

Options

A. 1
B. 7
C. 13
D. 17

Explanation

To solve the equation

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

we will use the property of inverse trigonometric functions.

### Step 1: Use the property of inverse sine and cosine
We know that

\[ \sin^{-1}(a) + \cos^{-1}(a) = \frac{\pi}{2}. \]

This means that

\[ \sin^{-1}\left(\frac{5}{x}\right) + \cos^{-1}\left(\frac{12}{x}\right) = \frac{\pi}{2}. \]

From this, we can deduce that

\[ \cos^{-1}\left(\frac{12}{x}\right) = \sin^{-1}\left(\frac{5}{x}\right). \]

### Step 2: Convert cosine inverse to sine inverse
Using the identity for cosine, we can express

\[ \cos^{-1}\left(\frac{12}{x}\right) = \sin^{-1}\left(\sqrt{1 - \left(\frac{12}{x}\right)^2}\right). \]

### Step 3: Set the arguments equal
Now, we set the two sine arguments equal to each other:

\[ \frac{5}{x} = \sqrt{1 - \left(\frac{12}{x}\right)^2}. \]

### Step 4: Square both sides
Squaring both sides gives:

\[ \left(\frac{5}{x}\right)^2 = 1 - \left(\frac{12}{x}\right)^2. \]

This simplifies to:

\[ \frac{25}{x^2} = 1 - \frac{144}{x^2}. \]

### Step 5: Combine the fractions
Now, combine the fractions:

\[ \frac{25}{x^2} + \frac{144}{x^2} = 1. \]

This leads to:

\[ \frac{169}{x^2} = 1. \]

### Step 6: Solve for \(x^2\)
Cross-multiplying gives:

\[ 169 = x^2. \]

### Step 7: Take the square root
Taking the square root of both sides results in:

\[ x = 13 \quad \text{(since \(x\) must be positive)}. \]

### Final Answer
Thus, the value of \(x\) is

\[ \boxed{13}. \]