Prove this identity: tan x = (sin (x + 2pi))/(cos(x - 2pi))

tan(x) = sin(x+2pi) /
cos(x-2pi)

We will use trigonometric identities to prove
the equality.

We know
that:

sin(a+ b) = sina*cosb+
cosa*sinb

==> sin(x+2pi) = sinx*cos2pi +
cosx*sin2pi

==> sin(x+2pi) = sinx*1 + cosx* 0 =
sinx,..........(1)

Now we know
that:

cos(a-b) = cosa*cosb-
sina*sinb

==> cos(x-2pi) = cosx*cos2pi +
sinx*sin2pi

                      = cosx*1 + sinx*0 =
cosx.............(2)

Now, from (1) and (2) we conclude
that:

sin(x+2pi)/cos(x-2pi) = sinx/cos x = tanx
.........q.e.d

==> Then we prove that
tanx = sin(x+2pi)/cos(x-2pi)