`\qquad \sqrt{1-({x_1x_2+y_1y_2}/{\sqrt{x_1^2+y_1^2}.\sqrt{x_2^2+y_2^2}})^2}`
`=\sqrt{1-{(x_1x_2+y_1y_2)^2}/{(x_1^2+y_1^2).(x_2^2+y_2^2)}`
`=\sqrt{{(x_1^2+y_1^2).(x_2^2+y_2^2)-(x_1^2x_2^2+2x_1x_2y_1y_2+y_1^2y_2^2)}/{(x_1^2+y_1^2).(x_2^2+y_2^2)}`
`=\sqrt{{x_1^2x_2^2+x_1^2y_2^2+y_1^2x_2^2+y_1^2y_2^2-x_1^2x_2^2-2x_1x_2y_1y_2-y_1^2y_2^2}/{(x_1^2+y_1^2).(x_2^2+y_2^2)}`
`=\sqrt{{x_1^2y_2^2+y_1^2x_2^2-2x_1x_2y_1y_2}/{(x_1^2+y_1^2).(x_2^2+y_2^2)}`
`=\sqrt{{(x_1y_2-x_2y_1)^2}/{(x_1^2+y_1^2).(x_2^2+y_2^2)}`
`=|x_1y_2-x_2y_1|/{\sqrt{x_1^2+y_1^2}.\sqrt{x_2^2+y_2^2}}`
