If `sec theta - cosec theta = 0` , then the value of `(sec theta + cosec theta)` is :

Options

A. `(sqrt3)/(2)`
B. `(2)/(sqrt3)`
C. 0
D. `2 sqrt2`

Answer

To solve the equation \( \sec \theta - \csc \theta = 0 \) and find the value of \( \sec \theta + \csc \theta \), we can follow these steps:

### Step 1: Set up the equation
Given:
\[ \sec \theta - \csc \theta = 0 \]
This implies:
\[ \sec \theta = \csc \theta \]

### Step 2: Rewrite secant and cosecant in terms of sine and cosine
Recall the definitions:
\[ \sec \theta = \frac{1}{\cos \theta} \quad \text{and} \quad \csc \theta = \frac{1}{\sin \theta} \]
Substituting these into the equation gives:
\[ \frac{1}{\cos \theta} = \frac{1}{\sin \theta} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying results in:
\[ \sin \theta = \cos \theta \]

### Step 4: Solve for theta
The equation \( \sin \theta = \cos \theta \) holds true when:
\[ \tan \theta = 1 \]
This occurs at:
\[ \theta = 45^\circ + n \cdot 180^\circ \quad (n \in \mathbb{Z}) \]

### Step 5: Calculate secant and cosecant at \( \theta = 45^\circ \)
Now, substituting \( \theta = 45^\circ \):
\[ \sec 45^\circ = \frac{1}{\cos 45^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]
\[ \csc 45^\circ = \frac{1}{\sin 45^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]

### Step 6: Find \( \sec \theta + \csc \theta \)
Now, we can find:
\[ \sec 45^\circ + \csc 45^\circ = \sqrt{2} + \sqrt{2} = 2\sqrt{2} \]

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