Is the product of two conjugate complex number always a real number?

Yes. This can be verified by using a "generic" complex number, and multiplying it by its conjugate:

(a + bi)(a - bi) = a2 -abi + abi + b2i2 = a2 - b2

Alternative proof:

I'm going to use the * notation for complex conjugate. Any complex number w is real if and only if w=w*. Let z be a complex number. Let w = zz*. We want to prove that w*=w. This is what we get:

w* = (zz*)*

= z*z** (for any u and v, (uv)* = u* v*)

= z*z

= w