Wykaż, że jeśli α jest kątem ostrym oraz sinα < ½, to cos²α × tg²α - cos²α < -½

  [latex]cos^2 alpha cdot tg^2 alpha - cos^2 alpha = \ =cos^2 alpha cdot frac{sin^2 alpha}{cos^2 alpha} - cos^2 alpha =\ =sin^2 alpha - cos^2 alpha =\ =sin^2 alpha -(1-sin^2 alpha) = \ =sin^2 alpha -1 + sin^2 alpha = \ =2sin^2 alpha -1\ sin alpha < frac{1}{2}\ sin^2alpha < frac{1}{4}\ 2sin^2 alpha -1 < 2cdot frac{1}{4} -1 \ 2sin^2 alpha -1 < frac{1}{2} -1\ 2sin^2 alpha -1 < -frac{1}{2}[/latex]