Question

Prove the identity
2 - (sinx + cosx)² = (sinx - cosx)²

Answer 1

Starting with the left side of the equation:
2 - (sin(x) + cos(x))^2

Expanding the square:
2 - (sin^2(x) + 2sin(x)cos(x) + cos^2(x))

Using the Pythagorean identity sin^2(x) + cos^2(x) = 1:
2 - (1 + 2sin(x)cos(x))

Simplifying:
2 - 1 - 2sin(x)cos(x)

Combining like terms:
1 - 2sin(x)cos(x)

Now, let's look at the right side of the equation:
(sin(x) - cos(x))^2

Expanding the square:
sin^2(x) - 2sin(x)cos(x) + cos^2(x)

Using the Pythagorean identity sin^2(x) + cos^2(x) = 1:
1 - 2sin(x)cos(x) + 1

Simplifying:
2 - 2sin(x)cos(x)

This matches the left side of the equation.

Therefore, the identity is proven:
2 - (sinx + cosx)^2 = (sinx - cosx)^2