Question

Упростите Докажите тождество

Answer 1

$1)\quad \frac{1-2cos^2a}{cosa+sina}=\frac{sin^2a+cos^2a-2cos^2a}{cosa+sina}=\frac{sin^2a-cos^2a}{cosa+sina}=\\\\=\frac{(sina-cosa)(sina+cosa)}{sina+cosa} =sina-cosa\\\\2)\quad \frac{sin^2a}{1+cosa}+cosa= \frac{sin^2a+cosa+cos^2a}{1+cosa}=\frac{1+cosa}{1+cosa} =1$

$3)\quad \frac{cos^2a-sin^2a}{cosa+sina}+tga\cdot cosa=\frac{(cosa-sina)(cosa+sina)}{cosa+sina} + \frac{sina}{cosa}\cdot cosa=\\\\=cosa-sina+sina=cosa\\\\cosa=cosa\\\\4)\quad \frac{(1-cos^2a)+tg^2a\cdot cos^2a}{sin^2a}=\frac{sun^2a+\frac{sin^2a}{cos^2a}\cdot cos^2a}{sin^2a}=\frac{sin^2a+sin^2a}{sin^2a} =\\\\=\frac{2sin^2a}{sin^2a}=2\\\\2=2$