Value of $A_k$ in Fourier series
I'm afraid that all the formulas in your question are inaccurate. From what I understand, you're interested in the representation of real-valued signals $x(t)$:

$$x(t)=a_0+2\Re\left\{\sum_{k=1}^{\infty}a_ke^{jk\omega_0t}\right\}=
a_0+2\sum_{k=1}^{\infty}\Re\left\{a_ke^{jk\omega_0t}\right\}\tag{1}$$

With $a_k=B_k+jC_k$ and $e^{jk\omega_0t}=\cos(k\omega_0t)+j\sin(k\omega_0t)$ you get

$$\Re\left\{a_ke^{jk\omega_0t}\right\}=B_k\cos(k\omega_0t)-C_k\sin(k\omega_0t)\tag{2}$$

which you can substitute into Eq. $(1)$ to get the equation in your question (but without the $j$!). The minus sign in $(2)$ appears simply because $j\cdot j=-1$.