If tan A = 1, then 2 sin A cos A = _______

Answer

To solve the problem "If tan A = 1, then 2 sin A cos A = _______", we can follow these steps:

### Step 1: Understand the value of tan A
Given that \( \tan A = 1 \), we know that this occurs when \( A = 45^\circ \) (or \( \frac{\pi}{4} \) radians), since \( \tan 45^\circ = 1 \).

**Hint:** Recall that \( \tan A = \frac{\text{opposite}}{\text{adjacent}} \) and find the angle where this ratio equals 1.

### Step 2: Use the double angle identity
We need to find the value of \( 2 \sin A \cos A \). We can use the double angle identity for sine:
\[ 2 \sin A \cos A = \sin(2A) \]

**Hint:** Remember the double angle formula for sine: \( \sin(2A) = 2 \sin A \cos A \).

### Step 3: Calculate \( 2A \)
Since \( A = 45^\circ \), we calculate:
\[ 2A = 2 \times 45^\circ = 90^\circ \]

**Hint:** Double the angle to find \( 2A \).

### Step 4: Find the sine of \( 90^\circ \)
Now we find:
\[ \sin(90^\circ) = 1 \]

**Hint:** Recall the values of sine for standard angles, especially \( \sin 90^\circ \).

### Step 5: Conclusion
Thus, we have:
\[ 2 \sin A \cos A = \sin(2A) = \sin(90^\circ) = 1 \]

**Final Answer:**
\[ 2 \sin A \cos A = 1 \]