If k = (1 + sin A) (1 + sinB) (1 + sin C) = (1 – sinA)(1 – sinB)(1 – sinC) then k =

$\begin{array}{rcl}\mathrm{k}& =& (1+\sin \mathrm{A}) (1+\sin \mathrm{B}) (1+\sin \mathrm{C})\\ & =& (1-\sin \mathrm{A}) (1-\sin \mathrm{B}) (1-\sin \mathrm{C})\\ {\mathrm{k}}^2& =& (1-{\sin }^2\mathrm{A}) (1-{\sin }^2\mathrm{B}) (1-{\sin }^2\mathrm{C})\\ {\mathrm{k}}^2& =& {\cos }^2\mathrm{A} {\cos }^2\mathrm{B} {\cos }^2\mathrm{C}\\ \mathrm{k}& =& \pm \mathrm{cosA} \mathrm{cosB} \mathrm{cosC}\end{array}$