If `sin^(-1)(x/13) + cosec^(-1) (13/12)= pi/2`, then the value of x is

Options

A. 5
B. 4
C. 12
D. 11

Answer

To solve the equation \( \sin^{-1}\left(\frac{x}{13}\right) + \csc^{-1}\left(\frac{13}{12}\right) = \frac{\pi}{2} \), we can follow these steps:

### Step 1: Rewrite the Cosecant Inverse
We know that \( \csc^{-1}(y) = \sin^{-1}\left(\frac{1}{y}\right) \). Therefore, we can rewrite \( \csc^{-1}\left(\frac{13}{12}\right) \) as:
\[ \csc^{-1}\left(\frac{13}{12}\right) = \sin^{-1}\left(\frac{12}{13}\right) \]

### Step 2: Substitute Back into the Equation
Now substitute this back into the original equation:
\[ \sin^{-1}\left(\frac{x}{13}\right) + \sin^{-1}\left(\frac{12}{13}\right) = \frac{\pi}{2} \]

### Step 3: Use the Identity
Using the identity \( \sin^{-1}(a) + \sin^{-1}(b) = \frac{\pi}{2} \) implies that:
\[ a + b = 1 \]
In our case:
\[ \frac{x}{13} + \frac{12}{13} = 1 \]

### Step 4: Solve for x
Now, we can solve for \( x \):
\[ \frac{x + 12}{13} = 1 \]
Multiplying both sides by 13 gives:
\[ x + 12 = 13 \]
Subtracting 12 from both sides results in:
\[ x = 1 \]

### Conclusion
Thus, the value of \( x \) is:
\[ \boxed{1} \]