On sait :
sin(2x) = 2cos(x)*sin(x)
cos(2x) = 1 - 2sin^2(x)
Tan(x) = sin(x) / cos(x)
Cos^2(x) + sin^2(x) = 1
Sin(2x) - tan(x) *cos(2x) =
2cos(x)*sin(x) - tan(x)* (1 - 2sin^2(x)) =
2cos(x)*sin(x) - sin(x)/cos(x) + 2sin^2(x)*tan(x) =
sin(x) [2cos(x) -1/cos(x) + 2sin(x)*sin(x)/cos(x)] =
sin(x)/cos(x) [2cos^2(x) -1 + 2sin^2(x)] =
tan(x) [2(cos^2(x) + sin^2(x)) - 1] =
tan(x) (2-1) = tan(x)
CQFD ;)
