Consider the equation
`sec theta +cosec theta=a, theta in (0, 2pi) -{pi//2, pi, 3pi//2}`
If the equation has no real roots, then

Options

A. `|a| ge 2sqrt(2)`
B. `a lt 2sqrt(2)`
C. `|a| lt 2sqrt(2)`
D. none of these

Explanation

To solve the equation \( \sec \theta + \csc \theta = a \) for \( \theta \in (0, 2\pi) - \{\frac{\pi}{2}, \pi, \frac{3\pi}{2}\} \) and determine the conditions under which it has no real roots, we can follow these steps:

### Step 1: Rewrite the equation in terms of sine and cosine
The secant and cosecant functions can be rewritten in terms of sine and cosine:
\[ \sec \theta = \frac{1}{\cos \theta}, \quad \csc \theta = \frac{1}{\sin \theta} \]
Thus, the equation becomes:
\[ \frac{1}{\cos \theta} + \frac{1}{\sin \theta} = a \]

### Step 2: Find a common denominator
To combine the fractions, we find a common denominator:
\[ \frac{\sin \theta + \cos \theta}{\sin \theta \cos \theta} = a \]

### Step 3: Rearranging the equation
Rearranging gives:
\[ \sin \theta + \cos \theta = a \sin \theta \cos \theta \]

### Step 4: Analyze the behavior of \( \sin \theta + \cos \theta \)
The maximum value of \( \sin \theta + \cos \theta \) can be found using the identity:
\[ \sin \theta + \cos \theta = \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) \]
The maximum value of \( \sin \) is 1, thus:
\[ \sin \theta + \cos \theta \leq \sqrt{2} \]

### Step 5: Analyze the behavior of \( a \sin \theta \cos \theta \)
The maximum value of \( \sin \theta \cos \theta \) occurs at \( \theta = \frac{\pi}{4} \) or \( \theta = \frac{3\pi}{4} \), where:
\[ \sin \theta \cos \theta = \frac{1}{2} \]
Thus, the maximum value of \( a \sin \theta \cos \theta \) is:
\[ \frac{a}{2} \]

### Step 6: Set up the condition for no real roots
For the equation \( \sin \theta + \cos \theta = a \sin \theta \cos \theta \) to have no real roots, the maximum value of \( \sin \theta + \cos \theta \) must be less than the minimum value of \( a \sin \theta \cos \theta \):
\[ \sqrt{2} < \frac{a}{2} \]
This can be rearranged to find the condition on \( a \):
\[ a > 2\sqrt{2} \]

### Conclusion
The condition for the equation \( \sec \theta + \csc \theta = a \) to have no real roots is:
\[ a > 2\sqrt{2} \]