Calculate the Product: Solve (x+y)² = 1 and (x²+y²)/(x+y)²=3

To solve the given problem, we will proceed with the following steps:

• Step 1: Apply the identity for the square of the sum.
• Step 2: Use the given equation to solve for the variables.
• Step 3: Derive the product $xy$.

Step 1: Using the identity
For $(x+y)^2 = x^2 + 2xy + y^2$, we know from the problem that $(x+y)^2 = 1$, so:

$x^2 + 2xy + y^2 = 1$

Step 2: Utilizing the second given equation,
We have $\frac{x^2+y^2}{(x+y)^2}=3$. Therefore:

$x^2 + y^2 = 3(x+y)^2 = 3 \times 1 = 3$

Step 3: Substitute $x^2+y^2 = 3$ into the identity:

$3 + 2xy = 1$

Solving for $xy$, we get:

$2xy = 1 - 3 = -2$

$xy = -1$

Therefore, the product of $x$ and $y$ is $\mathbf{-1}$.