Question

Помогите упростить выражения:
1. 1-cos^2a + sin^2a
2. sina / 1+cosa + 1+cosa/sina
3. (1/cos^2a-1) ctg^2a
4. sin^3a + cos^3a/ 1-sina cosa - cosa) ctga

Answer 1

$1. 1-cos^2a + sin^2a=sin^2a+sin^2a=2sin^2a\\ \\ \\ 2. \frac{sina}{1+cosa}+\frac{1+cosa}{sina} =\frac{sin^2a+1+2cosa+cos^2a}{(1+cosa)sina}=\frac{1+1+2cosa}{(1+cosa)*sina}=\frac{2(1+cosa)}{(1+cosa)*sina}=\frac{2}{sina} \\ \\ \\ 3. (\frac{1}{cos^2a}-1 )ctg^2a=\frac{1-cos^2a}{cos^2a}*ctg^2a=\frac{sin^2a}{cos^2a}*\frac{cos^2a}{sin^2a}$

$4. (\frac{sin^3a + cos^3a}{1-sina cosa} -cosa)*ctga=(\frac{(sina+cosa)(sin^2a-cosa*sina+sin^2a)}{1-sina*cosa} -cosa)*ctga=\\ \\(\frac{(sina+cosa)(1-cosa*sina)}{1-sina*cosa} -cosa)*ctga=(sina+cosa-cosa)*ctga=\\ \\ sina*\frac{cosa}{sina}=cosa$