Question

Tgx+ctgx/(tg^2x+1)(ctg^2x+1)

Answer 1

Не уверена что правильно

Answer 2

To simplify the expression:

First, let's express ctgx in terms of tgx:

ctgx = 1/tgx

Substituting this into the expression:

Tgx+ctgx/(tg^2x+1)(ctg^2x+1)

= Tgx + (1/tgx) / (tg^2x + 1)(1/tg^2x + 1)

= Tgx + tgx / (tg^2x + 1)(1 + tg^2x)

Now, let's use the identity:

1 + tg^2x = sec^2x

Substituting this into the expression:

Tgx + tgx / (tg^2x + 1)(1 + tg^2x)

= Tgx + tgx / (tg^2x + 1)sec^2x

= Tgx + tgx / (tg^2x + 1) (1/cos^2x)

= Tgx cos^2x + tgx sin^2x / (cos^2x)(tg^2x + 1)

= (Tgx cos^2x + sin^2x/cosx) / (tg^2x + 1)

= (Tgx cos^2x + 1/cosx) / (tg^2x + 1)

= (sinx/cosx cos^2x + 1/cosx) / (sin^2x/cos^2x + 1)

= (sinx + cosx) / (sin^2x + cos^2x)

= (sinx + cosx)/1

= sinx + cosx

Therefore, the simplified expression is sinx + cosx.