Coefficients of a sine series

Question

I am reading a paper on fluid mechanics- https://royalsocietypublishing.org/doi/10.1098/rspa.1954.0197 . Here, the author takes the sum of sines $\overline{WT}=\Sigma_{n=1}^\infty\Sigma_{m=1}^\infty\alpha_{nm}\sin{\frac{n\pi z}{d}}\sin{\frac{m\pi z}{d}}$, and equates it to $\overline{WT}=\Sigma_{n=1}^\infty(\alpha_{n}\sin^2{\frac{n\pi z}{d}}+\gamma_{n}\sin{\frac{n\pi z}{d}}\sin{\frac{(n-1)\pi z}{d}})$. He then relates the coefficients as shown in the image 1. I am not able to understand how these series were equated and how the values of the coefficients were obtained. Can someone please explain the math behind it?