

cosA+cosB+cosC+cos3π
=2.cos2A+B.cos2A−B+2.cos(2C+6π).cos(2C−6π)
≤2.cos2A+B+2.cos(2C+6π)=4.cos(12A+B+C+12π).cos(12A+B+C−12π)
≤4.cos(12A+B+C+12π)=4.cos3π
\Rightarrow
cosA+cosB+cosC+cos≤3.cos3π=23
dấu = xảy ra khi
A=B=C
cách khác
cosA+cosB+cosC≤23
\Leftrightarrow
2.cos2A+B.cos2A−B+1−2.sin22C+23≤0
\Leftrightarrow
−2.sin22C+2.sin2C.cos2A−B−21≤0
f(t)=−2t2+2t.cos2A−B−21
Δ′=cos22A−B−1≤0
\Rightarrow
f(t)≤0
\Rightarrow đpcm