sin^2(t)*cos^2(t)*[(sin^2(t)/cos^2(t))+(cos^2(t)/sin^2(t))+2)]=
=sin^2(t)*cos^2(t)*[(sin^4(t)+cos^4(t)+2cos^2(t)*sin^2(t))/sin^2(t)*cos^2(t)]=
=(cos^2(t)+sin^2(t))^2=1
sin^2t cos^2t (tg^2 t + ctg^2 t+ 2)= 1
для наглядности разбью на действия
1)
2 = 1 + 1
ctg^2 t+ 1 = 1 / sin^2 t
tg^2 + 1 = 1 / cos^2
2)
1 / cos^2 + 1 / sin^2 t = ( sin^2 t + cos^2) / (sin^2t * cos^2t) = 1 /(sin^2t * cos^2t)
3)
sin^2t * cos^2t * 1 /(sin^2t * cos^2t) = (sin^2t * cos^2t) / (sin^2t * cos^2t) = 1
sin^2(t)*cos^2(t)*[(sin^2(t)/cos^2(t))+(cos^2(t)/sin^2(t))+2)]=
=sin^2(t)*cos^2(t)*[(sin^4(t)+cos^4(t)+2cos^2(t)*sin^2(t))/sin^2(t)*cos^2(t)]=
=(cos^2(t)+sin^2(t))^2=1
sin^2t cos^2t (tg^2 t + ctg^2 t+ 2)= 1
для наглядности разбью на действия
1)
2 = 1 + 1
ctg^2 t+ 1 = 1 / sin^2 t
tg^2 + 1 = 1 / cos^2
2)
1 / cos^2 + 1 / sin^2 t = ( sin^2 t + cos^2) / (sin^2t * cos^2t) = 1 /(sin^2t * cos^2t)
3)
sin^2t * cos^2t * 1 /(sin^2t * cos^2t) = (sin^2t * cos^2t) / (sin^2t * cos^2t) = 1