The Cauchy-Reimann Conditions require that for a complex function to be analytic, then it must agree to the following equations
du/dx = dv/dy
du/dy = -dv/dx
The derivatives here are partial derivatives. The functions u and v are the real and imaginary parts of the complex function. First, we need to determine the real and imaginary parts of the complex function tan z.
Let z = x + yi.
tan z = tan (x + yi)
= (tan x + tan yi) / (1 - tan x tan yi)
Since tan yi = i tanh y,
tan z = (tan x + i tanh y) / (1 - i tan x tanh y)
Continuing, you can now represent tan z as
tan z = u(x, y) + i v(x, y).
You can now continue checking the equations.