${\left(1 + \mathrm{tan\alpha } \mathrm{tan\beta }\right)}^2 + {\left(\mathrm{tan\alpha } - \mathrm{tan\beta }\right)}^2=$

$\begin{array}{rcl} & & {\left(1 + \mathrm{tan\alpha } \mathrm{tan\beta }\right)}^2 + {\left(\mathrm{tan\alpha } - \mathrm{tan\beta }\right)}^2\\ & =& 1+{\tan }^2\mathrm{\alpha } {\tan }^2\mathrm{\beta }+2\mathrm{tan\alpha } \mathrm{tan\beta }+{\tan }^2\mathrm{\alpha }+{\tan }^2\mathrm{\beta }-2\mathrm{tan\alpha } \mathrm{tan\beta }\\ & =& 1+{\tan }^2\mathrm{\alpha }+{\tan }^2\mathrm{\beta }+{\tan }^2\mathrm{\alpha } {\tan }^2\mathrm{\beta }\\ & =& (1+{\tan }^2\mathrm{\alpha })(1+{\tan }^2\mathrm{\beta })\\ & =& {\sec }^2\mathrm{\alpha } {\sec }^2\mathrm{\beta }\end{array}$