\(cosA=2sin\frac{B+C}{2}.cos\frac{B-C}{2}-\frac{3}{2}\)
\(\Leftrightarrow cosA=2cos\frac{A}{2}.cos\frac{B-C}{2}-\frac{3}{2}\)
\(\Leftrightarrow2cos^2\frac{A}{2}-1=2cos\frac{A}{2}cos\frac{B-C}{2}-\frac{3}{2}\)
\(\Leftrightarrow4cos^2\frac{A}{2}-4cos\frac{A}{2}cos\frac{B-C}{2}+1=0\)
Mà \(\left\{{}\begin{matrix}cos\frac{A}{2}>0\\0< cos\frac{B-C}{2}\le1\end{matrix}\right.\) \(\Rightarrow cos\frac{A}{2}.cos\frac{B-C}{2}\le cos\frac{A}{2}\)
\(\Rightarrow4cos^2\frac{A}{2}-4cos\frac{A}{2}+1\le0\)
\(\Leftrightarrow\left(2cos\frac{A}{2}-1\right)^2\le0\)
BPT có nghiệm khi và chỉ khi:
\(\left\{{}\begin{matrix}2cos\frac{A}{2}-1=0\\B-C=0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}cos\frac{A}{2}=\frac{1}{2}\\B=C\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}A=120^0\\B=C\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}A=120^0\\B=C=30^0\end{matrix}\right.\)
