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

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### 1 Answer

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).**