If `cosec theta = a + (1)/( 4a)`, then the value of `cosec theta + cot theta ` is

Options

A. `-2a`
B. 2a
C. `- ( 1)/( 2a )`
D. `( 1)/( 2a)`

Answer

To find the value of \( \csc \theta + \cot \theta \) given that \( \csc \theta = a + \frac{1}{4a} \), we can follow these steps:

### Step 1: Express \( \cot \theta \) in terms of \( \csc \theta \)
We know that:
\[ \csc^2 \theta = 1 + \cot^2 \theta \]
From this, we can express \( \cot^2 \theta \) as:
\[ \cot^2 \theta = \csc^2 \theta - 1 \]

### Step 2: Calculate \( \csc^2 \theta \)
Since \( \csc \theta = a + \frac{1}{4a} \), we can find \( \csc^2 \theta \):
\[ \csc^2 \theta = \left(a + \frac{1}{4a}\right)^2 = a^2 + 2 \cdot a \cdot \frac{1}{4a} + \left(\frac{1}{4a}\right)^2 \]
\[ = a^2 + \frac{1}{2} + \frac{1}{16a^2} \]

### Step 3: Substitute \( \csc^2 \theta \) into \( \cot^2 \theta \)
Now we substitute \( \csc^2 \theta \) into the equation for \( \cot^2 \theta \):
\[ \cot^2 \theta = \left(a^2 + \frac{1}{2} + \frac{1}{16a^2}\right) - 1 \]
\[ = a^2 - \frac{1}{2} + \frac{1}{16a^2} \]

### Step 4: Find \( \cot \theta \)
Taking the square root to find \( \cot \theta \):
\[ \cot \theta = \sqrt{a^2 - \frac{1}{2} + \frac{1}{16a^2}} \]

### Step 5: Calculate \( \csc \theta + \cot \theta \)
Now we can find \( \csc \theta + \cot \theta \):
\[ \csc \theta + \cot \theta = \left(a + \frac{1}{4a}\right) + \sqrt{a^2 - \frac{1}{2} + \frac{1}{16a^2}} \]

### Final Expression
Thus, the value of \( \csc \theta + \cot \theta \) is:
\[ \csc \theta + \cot \theta = a + \frac{1}{4a} + \sqrt{a^2 - \frac{1}{2} + \frac{1}{16a^2}} \]