Question

$7^{tgx} + 7^{ctg( frac{ pi }{2}+x )} leq frac{50}{7}$

Answer 1

$7^{tgx}+7^{ctg(90+x)} leq frac{50}{7}\ 7^{tgx}+7^{-tgx} leq frac{50}{7}\ 7^{2tgx}+1 leq frac{50}{7}\ 7^{2tgx}*7+7 leq 50*7^{tgx}\ 7^{tgx}=y\ 7y^2-50y+7 leq 0\ D=2500-4*49=48^2\ y_{1}=frac{50+48}{14}=7\ y_{2}=frac{50-48}{14}=frac{1}{7}\ \ 7^{tgx}=7\ tgx=1\ x=frac{pi}{4}+pi*n\ \ 7^{tgx}=7^{-1}\ tgx=-1\ x=-frac{pi}{4}+pi*n$