[tex]\frac{1+sin^{2}x}{2+ctg^{2}x}+\frac{1+cos^{2}x}{2+tg^{2}x}=\frac{1+sin^{2}x}{2+\frac{cos^{2}x}{sin^2{x}}}+\frac{1+cos^{2}x}{2+\frac{sin^{2}x}{cos^{2}x}}[/tex][tex]=\frac{1+sin^{2}x}{\frac{2sin^{2}x+cos^{2}x}{sin^{2}x}}+\frac{1+cox^{2}x}{\frac{2cox^{2}x+sin^{2}x}{cox^{2}x}}=[/tex][tex]=\frac{1+sin^{2}x}{\frac{sin^{2}x+1}{sin^{2}x}}+\frac{1+cox^{2}x}{\frac{cox^{2}x+1}{cox^{2}x}}=[/tex][tex]sin^{2}x+cos^{2}x=1[/tex]
