Solve the system x-y=pi/6, tanx=tan2y.

We'll note the equations of the
system:

 x-y=pi/6
(1)

 tanx=tan2y (2)

We'll
write tan x = sin x/cos x

tan 2y = sin 2y/ cos
2y

We'll substitute tan x and tan 2y into the second
equation:

sin x/cos x = sin 2y/ cos
2y

We'll cross multiply and we'll
get:

sin x*cos 2y = cos x*sin
2y

We'll move all terms to one
side:

sin x*cos 2y - cos x*sin 2y =
0

sin (x - 2y) = 0

x - 2y =
arcsin 0

x - 2y = 0

x =
2y(3)

We'll form the system from the equations (1) and
(3):

 x - y=pi/6

x =
2y

We'll substitute x by 2y in
(1):

2y - y =
pi/6

y =
pi/6

x = 2y

x =
2pi/6

x =
pi/3

The solution of the
system is: (pi/3 ; pi/6).