cos$\alpha$cos$\beta$ = $\frac{x}{2}$ and sin$\alpha$sin$\beta$= $\frac{y}{2}$ ……………….(given)

sec(α-iβ) = $\frac{1}{cos⁡(α-iβ)}$ = $\frac{\frac{1}{(cosαcosiβ+sinαsiniβ)}}{(cosαcoshβ+isinαsinhβ)}$ $\frac{1}{(\frac{x}{2} +i\frac{y}{2})}$ =$\frac{2}{(x +iy)}$ …………………..(1)

sec(α+iβ) = $\frac{2}{(x-iy)}$ ……………….(2)

from (1) and (2)

sec(α-iβ) + sec(α-iβ-iβ) = $\frac{2}{(x +iy)} + \frac{2}{(x-iy)} = \frac{4x}{(x^2-y^2)}$