[tex](1+tgx)cos^{3}x+(1+ctgx)sin^{3}x=sinx+cosx[/tex]
[tex](1+\frac{sinx}{cosx} )cos^{3}x+(1+\frac{cosx}{sinx} )sin^{3}x=sinx+cosx[/tex]
[tex](cosx+sinx)cos^{2}x+(sinx+cosx)sin^{2}x=sinx+cosx[/tex]
[tex](cosx+sinx)cos^{2}x+(sinx+cosx)sin^{2}x=sinx+cosx |:(sinx+cosx)[/tex]
[tex]cos^{2}x+sin^{2}x=1[/tex]
