\(\large\frac{\sin{B}+\sin{C}}{\sin{A}}=\cot{\frac{A}{2}}\)
\(\large\frac{2.\sin{\frac{B+C}{2}}.\cos{\frac{B-C}{2}}}{sin{A}}=\cot{\frac{A}{2}}\)
\(\large 2.\frac{\sin\frac{180^\circ-A}{2}.\cos\frac{B-C}2}{\sin A}=\cot\frac{A}2\)
\(\large 2.\frac{\sin(90^\circ-\frac{A}2).\cos\frac{B-C}2}{\sin A}=\cot\frac{A}2\)
\(\large \frac{2.\cos{\frac{A}{2}}.\cos{\frac{B-C}{2}}}{2.\sin{\frac{A}{2}}.\cos{\frac{A}{2}}}=\frac{\cos{\frac{A}{2}}}{\sin{\frac{A}{2}}}\)
\(\large\cos{\frac{B-C}{2}}=\cos{\frac{A}{2}}\)
\(\frac{B-C}{2}=\frac{A}{2}\;\vee\;\frac{B-C}{2}=\frac{-A}{2}\)
\(\frac{B-C}{2}=\frac{A}{2}\;\vee\;\frac{B-C}{2}=\frac{-A}{2}\)
B − C = A   ∨ B − C = −A
B = A + C   ∨   A + B = C
B = 180° − B   ∨   180°− C = C
2B = 180°   ∨   180° = 2C
B = 90°   ∨  C = 90°