Question

Verify the following Pythagorean identity for all values of x and y : (x²+y²)²=(x²-y²)²+(2xy)²

Answer 1

To verify the Pythagorean identity (x²+y²)² = (x²-y²)² + (2xy)², both sides of the equation were expanded and shown to be equal through straightforward algebraic manipulation.

Step-by-step explanation:

The student's question involves verifying a Pythagorean identity. The identity to verify is (x²+y²)² = (x²-y²)² + (2xy)².

Let's break it down step-by-step:

1. Start by expanding the left-hand side (LHS) of the equation: (x²+y²)(x²+y²) = x´ + 2x²y² + y´.
2. Next, expand the first term on the right-hand side (RHS): (x²-y²)(x²-y²) = x´ - 2x²y² + y´.
3. Now, expand the second term on the RHS: (2xy)(2xy) = 4x²y².
4. Add the expanded terms of the RHS: x´ - 2x²y² + y´ + 4x²y² = x´ + 2x²y² + y´, which matches the expanded LHS.

Therefore, the identity is verified as both sides of the equation are equal.