Given 4cosx+2sinx=0 find tan2x

If we'll divide the constraint from enunciation by cos x,
the expression will become:

4 + 2sin x/cos x =
0

We'll subtract 4:

2sin x/cos
x = -4

sin x/cos x = -4/2

sin
x/cos x = -2

But sin x/cos x = tan
x

tan x = -2

We'll write tan
2x:

tan 2x=tan (x+x)

tan 2x =
(tan x+ tan x)/[1-(tan x)^2]

tan 2x =  2tan x/[1-(tan
x)^2]

We'll substitute tan x =
-2

tan 2x = 
2*(-2)/[1-(-2)^2]

tan 2x =
-4/(1-4)

tan 2x =
-4/-3

tan 2x =
4/3