Question

Упростите:

$frac{1}{ctgalpha}+frac{cosalpha}{1+sinalpha}$

Ответ $frac{1}{cosalpha}}$

Answer 1

Легко:
1/ctga=1/(cosa/sina)=sina/cosa.

Sina/cosa+cosa/(1+sina)=((sina*(1+sina)+cos^2(a))/(cosa+cosa*sina)=(sina+sin^2(a)+cos^2(a))/(cosa*(1+sina)=(sina+1)/(cosa*(1+sina))=1/cosa.

Answer 2

$frac{1}{ctgalpha} + frac{cosalpha}{1+sinalpha} = \ =tgalpha + frac{cosalpha}{1+sinalpha} = \ =frac{sinalpha}{cosalpha} + frac{cosalpha}{1+sinalpha} = \ =frac{sinalpha+sin^2alpha + cos^2alpha}{cosalpha(1+sinalpha)}=\ = frac{sinalpha+sin^2alpha + cos^2alpha}{cosalpha(1+sinalpha)} =\ = frac{1+sinalpha}{cosalpha(1+sinalpha)} = frac{1}{cosalpha}$