If `sin^(-1)"" x/5 + cosec^(-1)"" 5/4 = pi/2 " then " x=`

Options

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

Answer

To solve the equation \( \sin^{-1}\left(\frac{x}{5}\right) + \csc^{-1}\left(\frac{5}{4}\right) = \frac{\pi}{2} \), we can follow these steps:

### Step 1: Rewrite the equation
We know that \( \csc^{-1}(y) = \frac{\pi}{2} - \sin^{-1}\left(\frac{1}{y}\right) \). Therefore, we can rewrite the equation as:
\[ \sin^{-1}\left(\frac{x}{5}\right) + \left(\frac{\pi}{2} - \sin^{-1}\left(\frac{4}{5}\right)\right) = \frac{\pi}{2} \]

### Step 2: Simplify the equation
Now, we can simplify the equation:
\[ \sin^{-1}\left(\frac{x}{5}\right) - \sin^{-1}\left(\frac{4}{5}\right) = 0 \]

### Step 3: Set the inverse sine functions equal
From the equation above, we can deduce:
\[ \sin^{-1}\left(\frac{x}{5}\right) = \sin^{-1}\left(\frac{4}{5}\right) \]

### Step 4: Apply the sine function
Applying the sine function to both sides, we get:
\[ \frac{x}{5} = \frac{4}{5} \]

### Step 5: Solve for \( x \)
Now, we can solve for \( x \) by multiplying both sides by 5:
\[ x = 4 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{4} \]

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