# Find $\tan 2x$, given $\tan(x+y)=3$ and $\tan(x-y)=2$

I am having a hard time to solve this trigonometric system of equations. The equations is as follows:

We are given $$\tan(x+y)=3$$ $$\tan(x-y) = 2$$ and we need to find $$\tan2x$$

I have tried multiple ways of solving it but can't get the answer. I have tried angle addition and subtraction identities for tangent. Then I have also derived this equation $$\tan(x+y)-1=\tan(x-y)$$and tried it but without any success.

Any help will be appreciated!

And sorry if this is a very simple thing. I am just a new guy in the world of trigonometry.

HINT:

Let $A=x+y, B=x-y$

$\tan(A+B)=\dfrac{\tan A+\tan B}{1-\tan A\tan B}$

Alternatively, $$x+y=m\pi+\arctan3,x-y=n\pi+\arctan2$$ where $m,n$ are arbitrary integers.

$$\arctan3+\arctan2=\cdots=\dfrac{3\pi}4$$

• @AndréNicolas, Thanks for your observation. – lab bhattacharjee Jul 21 '16 at 6:32
• Thanks @lab bhattacharjee! First one is beautiful! – sehrob Jul 21 '16 at 6:42