If `sin^(-1) . (x)/(5) +cosec^(-1). (5/3)=pi/2`, then `x=`

Options

A. 3
B. 4
C. 5
D. 1

Answer

To solve the equation \( \sin^{-1} \left( \frac{x}{5} \right) + \csc^{-1} \left( \frac{5}{3} \right) = \frac{\pi}{2} \), we can follow these steps:

### Step 1: Convert \( \csc^{-1} \) to \( \sin^{-1} \)
We know that:
\[ \csc^{-1}(y) = \sin^{-1}\left(\frac{1}{y}\right) \]
Thus, we can rewrite \( \csc^{-1} \left( \frac{5}{3} \right) \) as:
\[ \csc^{-1} \left( \frac{5}{3} \right) = \sin^{-1} \left( \frac{3}{5} \right) \]

### Step 2: Substitute into the equation
Now substituting this back into the original equation, we have:
\[ \sin^{-1} \left( \frac{x}{5} \right) + \sin^{-1} \left( \frac{3}{5} \right) = \frac{\pi}{2} \]

### Step 3: Use the identity for \( \sin^{-1} \)
We know from trigonometric identities that:
\[ \sin^{-1}(a) + \sin^{-1}(b) = \frac{\pi}{2} \implies a + b = 1 \]
In our case, let:
\[ a = \frac{x}{5} \quad \text{and} \quad b = \frac{3}{5} \]
Thus, we can write:
\[ \frac{x}{5} + \frac{3}{5} = 1 \]

### Step 4: Solve for \( x \)
Now, we can solve for \( x \):
\[ \frac{x + 3}{5} = 1 \]
Multiplying both sides by 5 gives:
\[ x + 3 = 5 \]
Subtracting 3 from both sides results in:
\[ x = 5 - 3 = 2 \]

### Step 5: Final value of \( x \)
Thus, the value of \( x \) is:
\[ \boxed{2} \]