${\sin }^2 {12}^0+ {\sin }^2 {21}^0+ {\sin }^2 {39}^0+ {\sin }^2 {48}^0- {\sin }^2 9^0 - {\sin }^2 {18}^0 =$

$\begin{array}{rcl} & & {\sin }^2{12}^0+\sin {21}^0+{\sin }^2{39}^0-{\sin }^2{9}^0+{\sin }^2{48}^0-{\sin }^2{18}^0\\ & =& 1-({\cos }^2{21}^0-{\sin }^2{12}^0)+\sin {48}^0 \sin {30}^0+\sin {66}^0 \sin {30}^0\\ & =& 1-\cos {33}^0 \cos 9^0+\frac{1}{2}(\sin {66}^0+\sin {48}^0)\\ & =& 1-\cos {33}^0 \cos 9^0+\frac{1}{2} 2\sin {57}^0 \cos 9^0\\ & =& 1-\cos {33}^0 \cos 9^0+\cos {33}^0 \cos 9^0\\ & =& 1\end{array}$