`theta` is a positive acute angle and `sin theta-costheta=0` , then the value of `(sectheta+"cosec"theta)` is

Options

A. 2
B. `sqrt(2)`
C. `2sqrt(2)`
D. `3sqrt(2)`

Answer

To solve the problem step by step, we start with the given equation and find the value of \( \sec \theta + \csc \theta \).

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

### Step 2: Rearranging the equation
Rearranging the equation gives:
\[ \sin \theta = \cos \theta \]

### Step 3: Identify the angle
Since \( \theta \) is a positive acute angle, the only angle where sine and cosine are equal is:
\[ \theta = 45^\circ \]

### Step 4: Calculate \( \sec \theta \) and \( \csc \theta \)
Now we need to find \( \sec \theta \) and \( \csc \theta \) at \( \theta = 45^\circ \).

1. **Calculate \( \sec 45^\circ \)**:
\[ \sec 45^\circ = \frac{1}{\cos 45^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]

2. **Calculate \( \csc 45^\circ \)**:
\[ \csc 45^\circ = \frac{1}{\sin 45^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]

### Step 5: Add \( \sec \theta \) and \( \csc \theta \)
Now we can add the two results:
\[ \sec 45^\circ + \csc 45^\circ = \sqrt{2} + \sqrt{2} = 2\sqrt{2} \]

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

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