If `sin^(-1)(5/x)+sin^(-1)((12)/x)=pi/2,` then `x` is equal to `7/(13)` (b) `4/3` (c) `13` (d) `(13)/7`

Options

A. `(7)/(13)`
B. `(4)/(3)`
C. `13`
D. `(13)/(7)`

Explanation

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
Calculating the squares, we have:
\[ \frac{25}{x^2} + \frac{144}{x^2} = 1 \]
Combining the fractions gives:
\[ \frac{25 + 144}{x^2} = 1 \]
\[ \frac{169}{x^2} = 1 \]

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

### Step 4: Solve for \( x \)
Taking the square root of both sides:
\[ x = \pm 13 \]

### Step 5: Determine the valid solution
Since \( x \) must be positive (as it is in the denominator of the original equation), we take:
\[ x = 13 \]

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