Help prove an identity

(sin(x)-cos(x))(sin(x)+cos(x))
expand:
sin(x)^2-cos(x)*sin(x)+sin(x)*cos(x)-cos(x)^2
with: cos(x)^2+sin(x)^2=1 is: sin(x)^2=1-cos(x)^2
(1-cos(x)^2)-cos(x)*sin(x)+sin(x)*cos(x)-cos(x)^2
with: -cos(x)*sin(x)+sin(x)*cos(x) = 0
(1-cos(x)^2)-0-cos(x)^2
1-cos(x)^2-cos(x)^2
1-2*cos(x)^2

with: cos(x)^2+sin(x)^2=1
cos(x)^2+sin(x)^2-2*cos(x)^2
1*cos(x)^2+sin(x)^2-2*cos(x)^2
1*cos(x)^2-2*cos(x)^2+sin(x)^2
-cos(x)^2+sin(x)^2
with: cos(2x)=cos(x)^2-sin(x)^2 is: -cos(2x)=-cos(x)^2+sin(x)^2
-cos(x)^2+sin(x)^2
-cos(2x)