If `sin A = (2 m n)/( m^(2) + n^(2))` , What is the value of tan A ?

Options

A. `(2 m n)/( m^(2) + n^(2))`
B. ` (2 m n)/( m^(2) - n^(2))`
C. ` (m^(2) - n^(2))/( 2 mn)`
D. ` (m^(2) + n^(2))/( m^(2) - n^(2))`

Answer

To find the value of \( \tan A \) given that \( \sin A = \frac{2mn}{m^2 + n^2} \), we can follow these steps:

### Step 1: Identify the values of sine, hypotenuse, and perpendicular
From the given \( \sin A = \frac{2mn}{m^2 + n^2} \), we can identify:
- Perpendicular (opposite side) = \( 2mn \)
- Hypotenuse = \( m^2 + n^2 \)

### Step 2: Use the Pythagorean theorem to find the base
We know from the Pythagorean theorem that:
\[ \text{Hypotenuse}^2 = \text{Perpendicular}^2 + \text{Base}^2 \]
Substituting the known values:
\[ (m^2 + n^2)^2 = (2mn)^2 + \text{Base}^2 \]

### Step 3: Expand both sides
Expanding the left side:
\[ m^4 + 2m^2n^2 + n^4 = 4m^2n^2 + \text{Base}^2 \]

### Step 4: Rearranging the equation
Rearranging gives us:
\[ \text{Base}^2 = m^4 + n^4 + 2m^2n^2 - 4m^2n^2 \]
\[ \text{Base}^2 = m^4 + n^4 - 2m^2n^2 \]

### Step 5: Factor the right side
The expression \( m^4 + n^4 - 2m^2n^2 \) can be factored as:
\[ \text{Base}^2 = (m^2 - n^2)^2 \]
Thus, the base is:
\[ \text{Base} = m^2 - n^2 \]

### Step 6: Calculate \( \tan A \)
Now we can find \( \tan A \):
\[ \tan A = \frac{\text{Perpendicular}}{\text{Base}} = \frac{2mn}{m^2 - n^2} \]

### Final Answer
Thus, the value of \( \tan A \) is:
\[ \tan A = \frac{2mn}{m^2 - n^2} \]

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