LHS = \(\frac{a-b}{a+b}\)
By sine rule a = k sin A
B = k sin B
Hence,
\(\frac{a-b}{a+b}=\frac{k\,sin\,A-k\,sin\,B}{k\,sin\,A+k\,sin\,B}\)
\(=\frac{sin\,A-sin\,B}{sin\,A+sin\,B}\)
\(=\frac{2\,cos\frac{A+B}{2}\,sin\frac{A-B}{2}}{2\,sin\frac{A+B}{2}\,cos\frac{A-B}{2}}\)
\(=cot\frac{A+B}{2}\,tan\frac{A-B}{2}\)
\(=\frac{tan\frac{A-B}{2}}{tan\frac{A+B}{2}}\) = RHS
∴ LHS = RHS
