Why does the sign flip when substituting in Euler's formula

Question

Why does the sign flip when substituting in Euler's formula into the below

\begin{eqnarray} \lambda - 2 + \lambda^{-1} &=& C^2_x(e^{ip} - 2 + e^{-ip}) + C^2_y(e^{iq} - 2 + e^{-iq})\\ &=& 2C^2_x(1 - \cos p) + 2C^2_y(1 - \cos q) \end{eqnarray}

Using $e^{ip} = \cos p + i \sin p$ I got

\begin{eqnarray} \lambda - 2 + \lambda^{-1} &=& C^2_x(e^{ip} - 2 + e^{-ip}) + C^2_y(e^{iq} - 2 + e^{-iq})\\ &=& C^2_x(\cos p + i \sin p -2 + \cos p -i \sin p) + C^2_y(\cos q + i \sin q -2 + \cos q -i \sin q)\\ &=& C^2_x(2\cos p -2) + C^2_y(2\cos q -2)\\ &=& 2C^2_x(\cos p -1) + 2C^2_y(\cos q -1) \end{eqnarray}

What am I missing here?

Answer 1

The sign seems to flip because your first formula is wrong. We have $$\cos x = \frac{e^{ix}+e^{-ix}}{2}$$ and therefore $$e^{ip} - 2 + e^{-ip}= 2(\cos p -1)$$