Question

tg (-x) +1/1-ctg x ​

Answer 1

(tg(-x)+1)/(1-ctgx)=(1-tgx)/(1-ctgx)=(1-(sinx/cosx))/(1-(cosx/sinx))=

=((cosx-sinx)/cosx)/((sinx-cosx)/sinx)=-((sinx-cosx)/cosx)/((sinx-cosx)/sinx)=

=-sinx/cosx=-tgx.

Answer 2

$displaystyle frac{tg(-x) + 1}{1-ctg(x)}=frac{-tg(x)+1}{1-frac{1}{tg(x)}}=frac{-(tg(x)-1)cdot tg(x)}{1-tg(x)}=boxed{-tg(x)}$