2 tan x(sin x + cos x) = sec x(1 + sin 2x − cos 2x)

Question

prove this 

Answer 1

2tan(x)(sin(x)+cos(x))=2sin(x)/cos(x)*sin(x)+2sin(x)/cos(x)*cos(x)=

(1/cos(x))(2sin²(x)+sin(2x))=sec(x)(1-(1-2sin²(x))+sin(2x))=

=sec(x)(1-cos(2x)+sin(2x))=sec(x)(1+sin(2x)-cos(2x)).