Answer
Verify the following identity:
$(sin(x) sin(y) - cos(x) cos(y)) (sin(x) sin(y) + cos(x) cos(y)) = sin(y)^2 - cos(x)^2$
Work Step by Step
Verify the following identity:
$(sin(x) sin(y) - cos(x) cos(y)) (sin(x) sin(y) + cos(x) cos(y)) = sin(y)^2 - cos(x)^2$
$(sin(x) sin(y) - cos(x) cos(y)) (cos(x) cos(y) + sin(x) sin(y))$
$= sin(x)^2 sin(y)^2 - cos(x)^2 cos(y)^2$:
$sin(x)^2 sin(y)^2 - cos(x)^2 cos(y)^2 = ^?sin(y)^2 - cos(x)^2$
$cos(x)^2 = 1 - sin(x)^2$:
$sin(x)^2 sin(y)^2 - 1 - sin(x)^2 cos(y)^2 = ^?sin(y)^2 - cos(x)^2$
$cos(y)^2 = 1 - sin(y)^2$:
$sin(x)^2 sin(y)^2 - 1 - sin(y)^2 (1 - sin(x)^2) = ^?sin(y)^2 - cos(x)^2$
$-(1 - sin(x)^2) (1 - sin(y)^2) = -1 + sin(x)^2 + sin(y)^2 - sin(x)^2 sin(y)^2$:
$-1 + sin(x)^2 + sin(y)^2 - sin(x)^2 sin(y)^2 + sin(x)^2 sin(y)^2 = ^?sin(y)^2 - cos(x)^2$
$-1 + sin(x)^2 + sin(y)^2 - sin(x)^2 sin(y)^2 + sin(x)^2 sin(y)^2 = -1 + sin(x)^2 + sin(y)^2$:
$-1 + sin(x)^2 + sin(y)^2 = ^?sin(y)^2 - cos(x)^2$
$cos(x)^2 = 1 - sin(x)^2$:
$-1 + sin(x)^2 + sin(y)^2 = ^?sin(y)^2 - 1 - sin(x)^2$
$-(1 - sin(x)^2) = sin(x)^2 - 1$:
$-1 + sin(x)^2 + sin(y)^2 = ^?sin(x)^2 - 1 + sin(y)^2$
The left hand side and right hand side are identical, thus verifying the identity.
