Question

need to prove that (sin(x)+sin(3x))\cos(x)+cos(3x))=tan(2x)

Answer 1

sinx+sin3x=2sin(x+3x)/2cos(x-3x)/2
=2sin(4x)/2cos(-2x)/2
=2sin2xcos(-x) cos(-x)=cosx
=2sin2xcosx..............A
cosx+cos3x=2cos(x+3x)/2cos(x-3x)/2
=2cos(4x)/2cos(-2x)/2
=2cos2xcosx.............B
DIVIDING A by B
(sinx+sin3x)/cosx+cos3x=(2sin2xcosx)/2cos2xcosx
cancelling cosx by cosx & 2 by 2 gives
=sin2x/cos2x
=tan2x