Question

Aratati ca l tgx+ctgx l >=2, pentru orice x admisibil.
l inseamna modul :)

Answer 1

$|tg(x) + ctg(x)| \geq 2$

tg(x) + ctg(x)
$\frac{sin(x)}{cos(x)} + \frac{cos(x)}{sin(x)}$
$\frac{sin^2(x)+cos^2(x)}{sin(x)cos(x)}$
$\frac{1}{ \frac{1}{2}sin(2x) }$
$\frac{2}{sin(2x)}$
2 csc(2x)

$|csc(2x)| \geq 1$ pentru orice x, atunci
$|2csc(2x)| \geq 2$ => $|tg(x)+ctg(x)| \geq 2$