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

Options

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

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: Square the terms
Calculating the squares:
\[ \frac{25}{x^2} + \frac{144}{x^2} = 1 \]

### Step 3: Combine the fractions
Combine the fractions on the left side:
\[ \frac{25 + 144}{x^2} = 1 \]
This simplifies to:
\[ \frac{169}{x^2} = 1 \]

### Step 4: Cross-multiply to solve for \( x^2 \)
Cross-multiplying gives:
\[ 169 = x^2 \]

### Step 5: Solve for \( x \)
Taking the square root of both sides:
\[ x = \pm 13 \]

### Step 6: Determine the valid solution
Since \( x \) must be positive (as it is in the denominator of the sine functions), we take:
\[ x = 13 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{13} \]