If cosec A + sin A = `5(1)/5` , find the value of `"cosec"^2 A+ sin^2A`.

Answer

To solve the problem, we need to find the value of \( \csc^2 A + \sin^2 A \) given that \( \csc A + \sin A = 5 \frac{1}{5} \).

### Step-by-step Solution:

1. **Convert the Mixed Fraction:**
\[ \csc A + \sin A = 5 \frac{1}{5} = 5 + \frac{1}{5} = \frac{25 + 1}{5} = \frac{26}{5} \]

**Hint:** Convert mixed fractions to improper fractions for easier calculations.

2. **Square Both Sides:**
\[ \left( \csc A + \sin A \right)^2 = \left( \frac{26}{5} \right)^2 \]
This expands to:
\[ \csc^2 A + \sin^2 A + 2 \csc A \sin A = \frac{676}{25} \]

**Hint:** Remember the formula \( (a + b)^2 = a^2 + b^2 + 2ab \).

3. **Use the Identity:**
Since \( \sin A = \frac{1}{\csc A} \), we can substitute:
\[ 2 \csc A \sin A = 2 \csc A \cdot \frac{1}{\csc A} = 2 \]

**Hint:** Recall that sine and cosecant are reciprocal functions.

4. **Substitute Back:**
Now substitute \( 2 \) back into the equation:
\[ \csc^2 A + \sin^2 A + 2 = \frac{676}{25} \]

**Hint:** Isolate the term you need to find.

5. **Rearranging the Equation:**
\[ \csc^2 A + \sin^2 A = \frac{676}{25} - 2 \]

**Hint:** To subtract, convert \( 2 \) into a fraction with the same denominator:
\[ 2 = \frac{50}{25} \]

6. **Final Calculation:**
\[ \csc^2 A + \sin^2 A = \frac{676}{25} - \frac{50}{25} = \frac{626}{25} \]

**Hint:** When subtracting fractions, ensure the denominators are the same.

### Final Answer:
\[ \csc^2 A + \sin^2 A = \frac{626}{25} \]