If `cosec theta + cot theta= 2.5`, then `cosec theta`= ?

Options

A. `1.45`
B. `1.35`
C. `1.50`
D. `1.55`

Answer

To solve the equation \( \csc \theta + \cot \theta = 2.5 \) and find the value of \( \csc \theta \), we can follow these steps:

### Step 1: Rewrite the equation
We know that:
\[ \csc^2 \theta - \cot^2 \theta = 1 \]
This can be factored as:
\[ (\csc \theta + \cot \theta)(\csc \theta - \cot \theta) = 1 \]
Let \( x = \csc \theta \) and \( y = \cot \theta \). Thus, we have:
\[ (x + y)(x - y) = 1 \]

### Step 2: Substitute the known value
From the problem, we know:
\[ x + y = 2.5 \]
Substituting this into the equation gives us:
\[ (2.5)(x - y) = 1 \]

### Step 3: Solve for \( x - y \)
To find \( x - y \), we rearrange the equation:
\[ x - y = \frac{1}{2.5} = \frac{1}{\frac{5}{2}} = \frac{2}{5} \]

### Step 4: Set up the system of equations
Now we have two equations:
1. \( x + y = 2.5 \) (Equation 1)
2. \( x - y = \frac{2}{5} \) (Equation 2)

### Step 5: Add the two equations
Adding Equation 1 and Equation 2:
\[ (x + y) + (x - y) = 2.5 + \frac{2}{5} \]
This simplifies to:
\[ 2x = 2.5 + \frac{2}{5} \]

### Step 6: Convert \( 2.5 \) to a fraction
Convert \( 2.5 \) to a fraction:
\[ 2.5 = \frac{25}{10} = \frac{5}{2} \]
Now, we need a common denominator to add \( \frac{5}{2} \) and \( \frac{2}{5} \):
\[ \frac{5}{2} = \frac{25}{10}, \quad \frac{2}{5} = \frac{4}{10} \]
So,
\[ 2.5 + \frac{2}{5} = \frac{25}{10} + \frac{4}{10} = \frac{29}{10} \]

### Step 7: Solve for \( x \)
Now we have:
\[ 2x = \frac{29}{10} \]
Dividing both sides by 2:
\[ x = \frac{29}{20} \]

### Step 8: Conclusion
Thus, the value of \( \csc \theta \) is:
\[ \csc \theta = \frac{29}{20} = 1.45 \]