Question

упростить выражение
1/1+ctg^2 a + 1/1+tg^2 a

Answer 1

Способ 1. ctg²α = 1 / tg²α.

$displaystyle frac{1}{1+ ctg^2 alpha}+ frac{1}{1+tg^2 alpha}= frac{1}{1+ frac{1}{tg^2 alpha}}+frac{1}{1+ tg^2 alpha}= frac{1}{frac{tg^2 alpha +1}{tg^2 alpha}}+ frac{1}{1+ tg^2 alpha}=$

$displaystyle = frac{tg^2 alpha}{tg^2 alpha +1}+ frac{1}{1+ tg^2 alpha} = frac{tg^2 alpha +1}{tg^2 alpha +1}=1$

Ответ: 1.

Способ 2. sin²α+cos²α=1; tg²α=sin²α/cos²α; ctg²α=cos²α/sin²α.

$displaystyle frac{1}{1+ ctg^2 alpha}+ frac{1}{1+ tg^2 alpha}=frac{1}{1+ frac{cos^2 alpha}{sin^2 alpha}}+ frac{1}{1+ frac{sin^2 alpha}{cos^2 alpha}}=frac{1}{frac{sin^2 alpha+cos^ alpha}{sin^2 alpha}}+\ +frac{1}{frac{cos^2 alpha+sin^2 alpha}{cos^2 alpha}}=frac{sin^2 alpha}{sin^2 alpha+cos^2 alpha}+ frac{cos^2 alpha}{cos^2 alpha+sin^2 alpha}=sin^2 alpha+cos^2 alpha=1$

Ответ: 1.