If `"cosec" theta = sqrt5` , find the value of :
`2+1/(sin^2theta)-(cos^2theta)/(sin^2theta)`

Answer

To solve the problem, we need to find the value of the expression given that \( \csc \theta = \sqrt{5} \).

### Step-by-Step Solution:

1. **Understanding the given value**:
Since \( \csc \theta = \sqrt{5} \), we can find \( \sin \theta \):
\[ \csc \theta = \frac{1}{\sin \theta} \implies \sin \theta = \frac{1}{\sqrt{5}}. \]

2. **Finding \( \sin^2 \theta \)**:
\[ \sin^2 \theta = \left(\frac{1}{\sqrt{5}}\right)^2 = \frac{1}{5}. \]

3. **Finding \( \cos^2 \theta \)**:
Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{1}{5} = \frac{4}{5}. \]

4. **Substituting into the expression**:
We need to evaluate:
\[ 2 + \frac{1}{\sin^2 \theta} - \frac{\cos^2 \theta}{\sin^2 \theta}. \]
First, calculate \( \frac{1}{\sin^2 \theta} \):
\[ \frac{1}{\sin^2 \theta} = \frac{1}{\frac{1}{5}} = 5. \]
Next, calculate \( \frac{\cos^2 \theta}{\sin^2 \theta} \):
\[ \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\frac{4}{5}}{\frac{1}{5}} = 4. \]

5. **Putting it all together**:
Now substitute these values back into the expression:
\[ 2 + 5 - 4 = 3. \]

### Final Answer:
The value of the expression is \( 3 \).

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