Q.Show that $\dfrac{a+b}{c}=\dfrac{\cos\frac{A-B}{2}}{\sin\frac{C}{2}}$.

Sine Rule

The Sine Rule (also called the Law of Sines) is the first tool we reach for whenever a triangle is not right-angled and we still need to relate its sides to its angles. It states that in any triangle $ABC$ with sides $a,b,c$ opposite to angles $A,B,C$ respectively,

where $R$ is the radius of the circle that passes through all three vertices — the circumcircle of the triangle.

Why it is true

Draw the circumcircle of $\triangle ABC$ with centre $O$ and radius $R$. Draw the diameter through $B$, meeting the circle again at $D$, so $BD=2R$. Because $BD$ is a diameter, the angle $\angle BCD$ subtended by it at any point $C$ on the circle is a right angle ($90^\circ$). Also, $\angle BDC$ and $\angle BAC$ are angles subtended by the same arc $BC$, so $\angle BDC=\angle BAC=A$. Looking at the right triangle $BCD$, the side $BC=a$ is opposite the angle $\angle BDC=A$, and the hypotenuse is $BD=2R$. So

Repeating the same construction with a diameter through $A$ (or $C$) gives $\dfrac{b}{\sin B}=2R$ and $\dfrac{c}{\sin C}=2R$. All three ratios equal the same $2R$, which is what makes the rule so powerful — it ties every side/opposite-angle pair to one common constant fixed by the triangle's size.

Why it matters …