Question

Упростите выражение
$cotalpha-tanalpha$

Answer 1

$displaystyle cot(alpha)-tan(alpha)=frac{cos(alpha)}{sin(alpha)}-frac{sin(alpha)}{cos(alpha)}=frac{cos(alpha)}{sin(alpha)}frac{cos(alpha)}{cos(alpha)}-frac{sin(alpha)}{cos(alpha)}frac{sin(alpha)}{sin(alpha)}$
$displaystyle=frac{cos(alpha)^2}{sin(alpha)cos(alpha)}-frac{sin(alpha)^2}{sin(alpha)cos(alpha)}=frac{cos(alpha)^2-sin(alpha)^2}{frac122sin(alpha)cos(alpha)}=frac{cos(2alpha)}{frac12sin(2alpha)};$
$displaystylethereforecot(alpha)-tan(alpha)=boxed{2cot(2alpha)}phantom{.}.$