The maximum value of x that satisfies the equation `sin^(-1)((2sqrt(15))/(|x|))=cos^(-1)((14)/(|x|))` is

Answer

To solve the equation \( \sin^{-1}\left(\frac{2\sqrt{15}}{|x|}\right) = \cos^{-1}\left(\frac{14}{|x|}\right) \), we will follow these steps:

### Step 1: Rewrite the equation
We know that \( \sin^{-1}(y) + \cos^{-1}(y) = \frac{\pi}{2} \). Therefore, we can rewrite the equation as:
\[ \sin^{-1}\left(\frac{2\sqrt{15}}{|x|}\right) + \cos^{-1}\left(\frac{14}{|x|}\right) = \frac{\pi}{2} \]
This implies:
\[ \cos^{-1}\left(\frac{14}{|x|}\right) = \frac{\pi}{2} - \sin^{-1}\left(\frac{2\sqrt{15}}{|x|}\right) \]

### Step 2: Use the identity for cosine inverse
Using the identity \( \cos^{-1}(y) = \sin^{-1}(\sqrt{1 - y^2}) \), we can express the cosine term:
\[ \cos^{-1}\left(\frac{14}{|x|}\right) = \sin^{-1}\left(\sqrt{1 - \left(\frac{14}{|x|}\right)^2}\right) \]

### Step 3: Set the expressions equal
Now we set the two expressions equal:
\[ \sin^{-1}\left(\frac{2\sqrt{15}}{|x|}\right) = \sin^{-1}\left(\sqrt{1 - \left(\frac{14}{|x|}\right)^2}\right) \]

### Step 4: Eliminate the sine inverse
Since the sine inverse function is one-to-one in its range, we can equate the arguments:
\[ \frac{2\sqrt{15}}{|x|} = \sqrt{1 - \left(\frac{14}{|x|}\right)^2} \]

### Step 5: Square both sides
Squaring both sides gives:
\[ \left(\frac{2\sqrt{15}}{|x|}\right)^2 = 1 - \left(\frac{14}{|x|}\right)^2 \]
This simplifies to:
\[ \frac{60}{x^2} = 1 - \frac{196}{x^2} \]

### Step 6: Combine terms
Rearranging the equation, we have:
\[ \frac{60}{x^2} + \frac{196}{x^2} = 1 \]
\[ \frac{256}{x^2} = 1 \]

### Step 7: Solve for |x|
Multiplying both sides by \( x^2 \) gives:
\[ 256 = x^2 \]
Taking the square root:
\[ |x| = 16 \]

### Step 8: Determine the maximum value of x
Thus, \( x \) can be either \( 16 \) or \( -16 \). Since we are looking for the maximum value of \( x \):
\[ \text{Maximum value of } x = 16 \]

### Final Answer:
The maximum value of \( x \) that satisfies the equation is \( \boxed{16} \).