How to prove tan z is analytic using cauchy-riemann conditions ... ?

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.