LHS = a [cos C – cos B]
= \(a[2\,sin{\frac{B+C}{2}sin\frac{B-C}{2}}]\)
\(= a[2\,cos{\frac{A}{2}sin\frac{B-C}{2}}]\)
\(= 2k\,sin\,A[cos{\frac{A}{2}sin\frac{B-C}{2}}]\)
\(= 2k\,2\,sin\,\frac{A}{2}\,cos\frac{A}{2}[cos{\frac{A}{2}sin\frac{B-C}{2}}]\)
\(= 2k\,cos^2\frac{A}{2}[2\,sin{\frac{A}{2}sin\frac{B-C}{2}}]\)
\(= 2k\,cos^2\frac{A}{2}k[sin\{90°-(\frac{B+C}{2})\}sin\frac{B-C}{2}]\)
\(= 2\,cos^2\frac{A}{2}k[cos(\frac{B+C}{2})sin(\frac{B-C}{2})]\)
\(= 2\,cos^2\frac{A}{2}k[sin\,B\,-sin\,C]\)
\(= 2\,cos^2\frac{A}{2}[k\,sin\,B\,-k\,sin\,C]\)
\(= 2\,cos^2\frac{A}{2}[b \,-\,c]\) = RHS
∴ LHS = RHS
