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 start by using the identity that relates \(\sin^{-1}(y)\) and \(\cos^{-1}(y)\):

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

### Step 1: Simplify the right-hand side
Using the identity, we can rewrite the right-hand side:

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

Thus, the equation becomes:

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

### Step 2: Rearranging the equation
From the above equation, we can isolate one of the inverse sine terms:

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

### Step 3: Apply the sine function
Taking the sine of both sides:

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

Using the co-function identity \(\sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta)\):

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

### Step 4: Use the cosine identity
We know that:

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

Thus,

\[ \cos\left(\sin^{-1}\left(\frac{12}{x}\right)\right) = \sqrt{1 - \left(\frac{12}{x}\right)^2} = \sqrt{1 - \frac{144}{x^2}}. \]

### Step 5: Set up the equation
Now we can rewrite our equation:

\[ \frac{5}{x} = \sqrt{1 - \frac{144}{x^2}}. \]

### Step 6: Square both sides
Squaring both sides to eliminate the square root gives:

\[ \left(\frac{5}{x}\right)^2 = 1 - \frac{144}{x^2}. \]

This simplifies to:

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

### Step 7: Clear the fractions
Multiply through by \(x^2\) to eliminate the denominators:

\[ 25 = x^2 - 144. \]

### Step 8: Rearranging the equation
Rearranging gives:

\[ x^2 = 169. \]

### Step 9: Solve for \(x\)
Taking the square root of both sides:

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

### Step 10: Verify the solutions
We need to check which of these values satisfy the original equation.

1. **For \(x = 13\)**:
\[ \sin^{-1}\left(\frac{5}{13}\right) + \sin^{-1}\left(\frac{12}{13}\right) = \frac{\pi}{2}. \]
This holds true.

2. **For \(x = -13\)**:
\[ \sin^{-1}\left(\frac{5}{-13}\right) + \sin^{-1}\left(\frac{12}{-13}\right) = -\frac{\pi}{2} \quad \text{(not equal to } \frac{\pi}{2}). \]
This does not hold true.

Thus, the only valid solution is:

\[ \boxed{13}. \]