Find the value of `sqrt(cosec^(2) 45^(@) - cot^(2) 45^(@))`

Answer

To solve the problem, we need to find the value of \( \sqrt{\csc^2 45^\circ - \cot^2 45^\circ} \).

### Step-by-Step Solution:

1. **Identify the Trigonometric Values**:
- We need to find the values of \( \csc^2 45^\circ \) and \( \cot^2 45^\circ \).
- \( \csc 45^\circ = \frac{1}{\sin 45^\circ} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \)
- Therefore, \( \csc^2 45^\circ = (\sqrt{2})^2 = 2 \).
- \( \cot 45^\circ = \frac{\cos 45^\circ}{\sin 45^\circ} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 \)
- Thus, \( \cot^2 45^\circ = 1^2 = 1 \).

2. **Substitute the Values**:
- Now substitute these values into the expression:
\[ \sqrt{\csc^2 45^\circ - \cot^2 45^\circ} = \sqrt{2 - 1} \]

3. **Simplify the Expression**:
- Simplifying the expression inside the square root:
\[ \sqrt{2 - 1} = \sqrt{1} \]

4. **Find the Final Value**:
- The square root of 1 is:
\[ \sqrt{1} = 1 \]

### Final Answer:
Thus, the value of \( \sqrt{\csc^2 45^\circ - \cot^2 45^\circ} \) is \( 1 \).

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