In any triangle `A B C` , prove that following :`(a+b)/c=(cos((A-B)/2))/(sin(C/2))`

Explanation

We have,



`= frac{{a}+{b}}{{C}} `


`= frac{2 {R}(sin {A}+sin {B})}{2 {R} sin {C}} text { [Using sine rule of triangle] } `


`= frac{sin {A}+sin {B}}{sin {C}} `


`= frac{2 cdot sin frac{{A}+{B}}{2} cdot cos frac{{A}-{B}}{2}}{2 cdot sin frac{{C}}{2} cdot cos frac{{C}}{2}} `


`= frac{sin (frac{pi}{2}-frac{{C}}{2}) cdot cos frac{{A}-{B}}{2}}{sin frac{{C}}{2} cdot cos frac{{C}}{2}}``[text { Since } {A}+{B}+{C}=180^{circ}]`



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