Question about cosine and Eulers identity

[Rib5]

I was doing a signals and systems problem and I think I might be screwing something up with the cosine function because I get

$$cos(a+b) = cos(a)*cos(b)$$

This is how

$$cos(a+b)=Re\left\{ e^{j*(a+b)} \right\}$$
$$=Re\left\{ e^{j*(a)}*e^{j*(b)} \right\}$$
$$=cos(a)*cos(b)$$

Can anyone point out my mistake?

EDIT:
I guess I figured it out. The Re{} function isn't distributive. Thanks for your time.

 

[snipez90]

The expression at the second to last line will involve a multiplication of two factors each having one j (imaginary unit), which results in the missing real expression.

 

[g_edgar]

The expression at the second to last line will involve a multiplication of two factors each having one j (imaginary unit), which results in the missing real expression.

In simpler language... the real part of a product is not the product of the real parts