Question

Докажите тождество
$frac{ sin^{2}a }{ tg^{2}a } + frac{ cos^{2}a }{ ctg^{2}a } - frac{1}{ sin^{2}a } = - ctg^{2}a$

$frac{cos^{3}a + sin^{3}a }{cosa + sina} + cosa * sina = 1$

Answer 1

$1)quad frac{sin^2a}{tg^2a} +frac{cos^2a}{ctg^2a}- frac{1}{sin^2a} = frac{sin^2a}{frac{sin^2a}{cos^2a}} + frac{cos^2a}{frac{cos^2a}{sin^2a}}-frac{1}{sin^2a} =\\=underbrace {cos^2a+sin^2a}_{1}-(1+ctg^2a)=1-1-ctg^2a=-ctg^2a\\2)quad frac{cos^3a+sin^3a}{cosa+sina} +cosacdot sina=\\= frac{(cosa+sina)(cos^2a-sinacdot cosa+sin^2a)}{cosa+sina} +cosacdot sina=\\= underbrace {cos^2a+sin^2a}_{1}-sinacdot cosa+cosacdot sina=1$