If `"cosec"theta+sintheta=(5)/(2)` , then the value of `("cosec"theta-sintheta)`is :

Options

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

Answer

To solve the equation \( \csc \theta + \sin \theta = \frac{5}{2} \) and find the value of \( \csc \theta - \sin \theta \), we can follow these steps:

### Step 1: Rewrite the equation
We know that \( \csc \theta = \frac{1}{\sin \theta} \). Therefore, we can rewrite the original equation as:
\[ \frac{1}{\sin \theta} + \sin \theta = \frac{5}{2} \]

### Step 2: Multiply through by \( \sin \theta \)
To eliminate the fraction, multiply every term by \( \sin \theta \):
\[ 1 + \sin^2 \theta = \frac{5}{2} \sin \theta \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ \sin^2 \theta - \frac{5}{2} \sin \theta + 1 = 0 \]

### Step 4: Use the quadratic formula
This is a quadratic equation in the form \( a\sin^2 \theta + b\sin \theta + c = 0 \). Here, \( a = 1 \), \( b = -\frac{5}{2} \), and \( c = 1 \). We can use the quadratic formula:
\[ \sin \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values:
\[ \sin \theta = \frac{\frac{5}{2} \pm \sqrt{\left(-\frac{5}{2}\right)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ = \frac{\frac{5}{2} \pm \sqrt{\frac{25}{4} - 4}}{2} \]
\[ = \frac{\frac{5}{2} \pm \sqrt{\frac{25}{4} - \frac{16}{4}}}{2} \]
\[ = \frac{\frac{5}{2} \pm \sqrt{\frac{9}{4}}}{2} \]
\[ = \frac{\frac{5}{2} \pm \frac{3}{2}}{2} \]

### Step 5: Calculate the two possible values for \( \sin \theta \)
Calculating the two cases:
1. \( \sin \theta = \frac{5 + 3}{4} = \frac{8}{4} = 2 \) (not possible since \( \sin \theta \) cannot exceed 1)
2. \( \sin \theta = \frac{5 - 3}{4} = \frac{2}{4} = \frac{1}{2} \)

### Step 6: Find \( \csc \theta \)
Now that we have \( \sin \theta = \frac{1}{2} \), we can find \( \csc \theta \):
\[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{1}{2}} = 2 \]

### Step 7: Calculate \( \csc \theta - \sin \theta \)
Now we can find \( \csc \theta - \sin \theta \):
\[ \csc \theta - \sin \theta = 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \]

### Final Answer
Thus, the value of \( \csc \theta - \sin \theta \) is:
\[ \frac{3}{2} \]