Skorzystamy ze wzorów na tangens i cotangens sumy kątów: [latex]tg (alpha + eta) = frac{tg alpha + tg eta}{1- tg alpha cdot tg eta} \\ ctg (alpha + eta) = frac{ctg alpha cdot ctg eta - 1}{ctg alpha + ctg eta}[/latex] [latex]tg^2120^o +2tg120^octg150^o +ctg^2150^o =(tg 120^o +ctg150^o)^2 = \\ = [tg(60^o +60^o) + ctg(90^o+60^o)]^2=(frac{tg60^o+tg60^o}{1 - tg60^o cdot tg60^o} + frac{ctg 90^o cdot ctg 60^o-1}{ctg 90^o + ctg 60^o})^2\\ = (frac{sqrt{3}+sqrt{3}}{1 - sqrt{3} cdot sqrt{3}} +frac{0 cdot frac{sqrt{3}}{3} -1}{0 +frac{sqrt{3}}{3}})^2 = (frac{2sqrt{3}}{1 - 3} +frac{-1}{frac{sqrt{3}}{3}})^2 = (frac{2sqrt{3}}{-2} -frac{3}{sqrt{3}})^2 = \\ =(- sqrt{3} - sqrt{3})^2 = (- 2sqrt{3})^2=12[/latex]
