I'm reading a paper (Beamwidth and directivity of large scanning arrays, R. S. Elliott, Appendix A) in which the author starts from this expression:
$$\frac{\sin\left [ \left ( 2N+1 \right ) u_0\right ]}{\sin(u_0)}\sum_{p=-P}^Pa_p\cos(p\pi)\left [\frac{\sin(u_0)}{\sin(u_p)} -1+1 \right ]$$
where $a_p=a_{-p}$ and $u_p=\frac{\pi L}{(N+1)\lambda}\left(\cos \theta'-\cos\theta_0+\frac{p\lambda}{L}\right)$, and says that we can approximate it in the following way:
$$( 2N+1)\frac{\sin\left [ \left ( 2N+1 \right ) u_0\right ]}{( 2N+1)u_0}\left \{ a_0+\sum_{p=1}^P2a_p\cos(p\pi) \right \} - ( 2N+1)\frac{\sin\left [ \left ( 2N+1 \right ) u_0\right ]}{( 2N+1)u_0}\sum_{p=1}^P2a_p\cos(p\pi)\frac{p^2}{p^2-K^2}$$
where $K=\frac{L}{\lambda}\left(\cos\theta'-\cos\theta_0\right)$, because "$u_0$ and $u_p$ are small and $P$ is a small integer".
I simply can't understand why. My attempt, using $\sin u_p\approx u_p$, gives the same approximate expression, but with the difference that I have a factor $\frac{p}{p+K}$ instead of the factor $\frac{p^2}{p^2-K^2}$.
What am I missing?