a) $m=1 $ => $P=\frac{2\cos x+2}{\sin x+\cos x+2}$
$<=> P\sin x+(P-2)\cos x=2-2P$ (*)
PT (*) có nghiệm khi: $P^{2}+(P-2)^{2}\geq (2-2P)^{2}$
$<=> (P-1)^{2}\leq 1$
=> $0\leq P \leq 2$
...
b) $P\sin x+cosx(P-2m)=m+1-2P$ (**)
PT (**) có nghiệm khi:
$P^{2}+(P-2m)^{2}\geq (m+1-2P)^{2}$.
...
$<=> 2P^{2}-3m^{2}-4P+2m+1\leq 0$
$Delta = 6(m-\frac{1}{3})^{2}+\frac{4}{3}\geq \frac{4}{3}$
=> PT luôn có nghiệm.
Min Delta $= \frac{4}{3}$ khi $m=1/3$
=> $Max_{P} min <=> m=\frac{1}{3}$