If `sin x - cos x = 0, 0^(@) lt x lt 90^(@)` then the value of `( sec x + cosec x)^(2)` is:

Options

A. 8
B. 4
C. 10
D. 6

Answer

To solve the problem step by step, we start with the given equation and find the required value.

### Step 1: Start with the given equation
We have:
\[ \sin x - \cos x = 0 \]

### Step 2: Rearrange the equation
Rearranging gives us:
\[ \sin x = \cos x \]

### Step 3: Divide both sides by \(\cos x\)
Dividing both sides by \(\cos x\) (since \(\cos x \neq 0\) in the interval \(0 < x < 90^\circ\)):
\[ \frac{\sin x}{\cos x} = 1 \]
This simplifies to:
\[ \tan x = 1 \]

### Step 4: Find the angle \(x\)
The angle \(x\) for which \(\tan x = 1\) in the interval \(0 < x < 90^\circ\) is:
\[ x = 45^\circ \]

### Step 5: Calculate \(\sec x\) and \(\csc x\)
Now we need to find \(\sec x\) and \(\csc x\) at \(x = 45^\circ\):
\[ \sec 45^\circ = \frac{1}{\cos 45^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]
\[ \csc 45^\circ = \frac{1}{\sin 45^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]

### Step 6: Calculate \((\sec x + \csc x)^2\)
Now we can find:
\[ \sec x + \csc x = \sqrt{2} + \sqrt{2} = 2\sqrt{2} \]
Now squaring this value:
\[ (\sec x + \csc x)^2 = (2\sqrt{2})^2 = 4 \cdot 2 = 8 \]

### Final Answer
Thus, the value of \((\sec x + \csc x)^2\) is:
\[ \boxed{8} \]