If `cosec theta-cottheta=(7)/(2)` , the value of `cosectheta` is :

Options

A. `(47)/(28)`
B. `(51)/(28)`
C. `(53)/(28)`
D. `(49)/(28)`

Answer

To solve the equation \( \csc \theta - \cot \theta = \frac{7}{2} \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \csc \theta - \cot \theta = \frac{7}{2} \]

### Step 2: Express \(\cot \theta\) in terms of \(\csc \theta\)
Recall that:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{1}{\tan \theta} = \frac{\csc \theta}{\sin \theta} \]
Thus, we can rewrite \(\cot \theta\) as:
\[ \cot \theta = \frac{\sqrt{\csc^2 \theta - 1}}{\csc \theta} \]

### Step 3: Substitute \(\cot \theta\) into the equation
Substituting this into the equation gives:
\[ \csc \theta - \frac{\sqrt{\csc^2 \theta - 1}}{\csc \theta} = \frac{7}{2} \]

### Step 4: Multiply through by \(\csc \theta\)
To eliminate the fraction, multiply the entire equation by \(\csc \theta\):
\[ \csc^2 \theta - \sqrt{\csc^2 \theta - 1} = \frac{7}{2} \csc \theta \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ \csc^2 \theta - \frac{7}{2} \csc \theta - \sqrt{\csc^2 \theta - 1} = 0 \]

### Step 6: Square both sides
Square both sides to eliminate the square root:
\[ \left(\csc^2 \theta - \frac{7}{2} \csc \theta\right)^2 = \csc^2 \theta - 1 \]

### Step 7: Expand and simplify
Expanding the left side:
\[ \csc^4 \theta - 7 \csc^3 \theta + \frac{49}{4} \csc^2 \theta = \csc^2 \theta - 1 \]
Rearranging gives:
\[ \csc^4 \theta - 7 \csc^3 \theta + \left(\frac{49}{4} - 1\right) \csc^2 \theta + 1 = 0 \]
This simplifies to:
\[ \csc^4 \theta - 7 \csc^3 \theta + \frac{45}{4} \csc^2 \theta + 1 = 0 \]

### Step 8: Solve the quadratic equation
Let \( x = \csc^2 \theta \):
\[ x^2 - 7x + \frac{45}{4} = 0 \]
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 1 \cdot \frac{45}{4}}}{2 \cdot 1} \]
\[ = \frac{7 \pm \sqrt{49 - 45}}{2} = \frac{7 \pm 2}{2} \]
Thus, \( x = \frac{9}{2} \) or \( x = \frac{5}{2} \).

### Step 9: Find \(\csc \theta\)
Since \( x = \csc^2 \theta \), we have:
\[ \csc^2 \theta = \frac{9}{2} \quad \text{or} \quad \csc^2 \theta = \frac{5}{2} \]
Taking the square root gives:
\[ \csc \theta = \frac{3}{\sqrt{2}} \quad \text{or} \quad \csc \theta = \frac{\sqrt{5}}{\sqrt{2}} \]

### Final Answer
The value of \(\csc \theta\) can be either:
\[ \csc \theta = \frac{3}{\sqrt{2}} \quad \text{or} \quad \csc \theta = \frac{\sqrt{5}}{\sqrt{2}} \]