In a right triangle ABC right-angled at B. if `t a n A=1`, then verify that `2sinA cos A=1.`

Options

A. `0`
B. `1`
C. `2`
D. `3`

Explanation

To solve the problem, we need to verify that \(2 \sin A \cos A = 1\) given that \(\tan A = 1\) in a right triangle ABC, where angle B is the right angle.

### Step-by-step Solution:

1. **Understanding the Given Information**:
We know that in triangle ABC, angle B is 90 degrees. Therefore, angles A and C must be acute angles that add up to 90 degrees.

2. **Using the Tangent Function**:
Given that \(\tan A = 1\), we can interpret this in terms of the definition of tangent in a right triangle:
\[ \tan A = \frac{\text{opposite}}{\text{adjacent}} = 1 \]
This implies that the lengths of the opposite side and the adjacent side are equal. Thus, we can conclude that angle A must be 45 degrees.

3. **Finding the Value of Angle A**:
Since \(\tan A = 1\), we can find that:
\[ A = 45^\circ \]

4. **Substituting into the Expression**:
Now we need to verify the expression \(2 \sin A \cos A = 1\). We substitute \(A = 45^\circ\):
\[ 2 \sin 45^\circ \cos 45^\circ \]

5. **Calculating \(\sin 45^\circ\) and \(\cos 45^\circ\)**:
We know that:
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \quad \text{and} \quad \cos 45^\circ = \frac{1}{\sqrt{2}} \]

6. **Substituting the Values**:
Now we substitute these values into the expression:
\[ 2 \sin 45^\circ \cos 45^\circ = 2 \left(\frac{1}{\sqrt{2}}\right) \left(\frac{1}{\sqrt{2}}\right) \]

7. **Simplifying the Expression**:
This simplifies to:
\[ 2 \cdot \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} = 2 \cdot \frac{1}{2} = 1 \]

8. **Conclusion**:
Therefore, we have verified that:
\[ 2 \sin A \cos A = 1 \]