If `sin^(-1)(x/5) + cosec^(-1) (5/4) = pi/2 `, then the value of x is

Options

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

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 using the identity
We know that:
\[ \sin^{-1}(a) + \csc^{-1}(b) = \frac{\pi}{2} \text{ if } a = \frac{1}{b} \]
In our case, we can rewrite the equation as:
\[ \sin^{-1}\left(\frac{x}{5}\right) + \sin^{-1}\left(\frac{4}{5}\right) = \frac{\pi}{2} \]

### Step 2: Set up the equation
Let:
\[ \theta = \sin^{-1}\left(\frac{4}{5}\right) \]
Then we can express the equation as:
\[ \sin^{-1}\left(\frac{x}{5}\right) + \theta = \frac{\pi}{2} \]

### Step 3: Isolate \( \sin^{-1}\left(\frac{x}{5}\right) \)
Rearranging gives us:
\[ \sin^{-1}\left(\frac{x}{5}\right) = \frac{\pi}{2} - \theta \]

### Step 4: Apply the sine function
Taking the sine of both sides:
\[ \sin\left(\sin^{-1}\left(\frac{x}{5}\right)\right) = \sin\left(\frac{\pi}{2} - \theta\right) \]
This simplifies to:
\[ \frac{x}{5} = \cos(\theta) \]

### Step 5: Find \( \cos(\theta) \)
From the right triangle definition, since \( \sin(\theta) = \frac{4}{5} \), we can find \( \cos(\theta) \) using the Pythagorean theorem:
\[ \cos^2(\theta) + \sin^2(\theta) = 1 \]
\[ \cos^2(\theta) + \left(\frac{4}{5}\right)^2 = 1 \]
\[ \cos^2(\theta) + \frac{16}{25} = 1 \]
\[ \cos^2(\theta) = 1 - \frac{16}{25} = \frac{9}{25} \]
\[ \cos(\theta) = \frac{3}{5} \]

### Step 6: Substitute back to find \( x \)
Substituting back into the equation:
\[ \frac{x}{5} = \frac{3}{5} \]
Multiplying both sides by 5 gives:
\[ x = 3 \]

### Final Answer
Thus, the value of \( x \) is \( 3 \).

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