sin^2t cos^2t(tg^2 t + ctg^2 t+ 2) Упростите выражение.

Question

sin^2t cos^2t(tg^2 t + ctg^2 t+ 2) Упростите выражение.

Answer 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

Answer 2

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