If `sin^(-1)((5)/(x))+sin^(-1)((12)/(x))=sin^(-1)((2)/(x))+cos^(-1)((2)/(x))`
then the value of x is equal to

Answer

To solve the equation

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

we can follow these steps:

### Step 1: Use the identity for inverse sine and cosine
We know that

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

Thus, we can rewrite the right-hand side:

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

### Step 2: Substitute into the equation
Substituting this into our equation gives:

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

### Step 3: Rearrange the equation
Rearranging the equation, we have:

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

### Step 4: Use the sine identity
Using the identity \(\sin^{-1}(a) = \frac{\pi}{2} - \cos^{-1}(a)\), we can express this as:

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

### Step 5: Apply the cosine identity
Using the identity \(\cos(\sin^{-1}(y)) = \sqrt{1 - y^2}\), we can write:

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

### Step 6: 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 7: Combine terms
Combining the terms leads to:

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

which simplifies to:

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

### Step 8: Solve for \(x^2\)
Cross-multiplying gives:

\[ 169 = x^2. \]

### Step 9: Take the square root
Taking the square root of both sides gives:

\[ x = 13 \quad \text{or} \quad x = -13. \]

### Step 10: Check for validity
We need to check which values are valid in the context of the original equation.

1. For \(x = 13\):
- \(\sin^{-1}\left(\frac{5}{13}\right) + \sin^{-1}\left(\frac{12}{13}\right) = \frac{\pi}{2}\) is valid.

2. For \(x = -13\):
- \(\sin^{-1}\left(\frac{-5}{13}\right) + \sin^{-1}\left(\frac{-12}{13}\right) \neq \frac{\pi}{2}\) (as it will yield \(-\frac{\pi}{2}\)).

Thus, the only valid solution is:

\[ \boxed{13}. \]