Question

CosA + cosB - cosC = -1 + 4cosA/2 cosB/2 sinC/2

Answer 1

Explanation:

(cos A+ cos B)-cos C

`=2cos (A+B)/(2)cos (A-B)/(2)-cos C~~~...(1)\\A+B+C=180\\A+B=180-C\\(A+B)/(2)=90-(C)/(2)\\cos (A+B)/(2)=cos(90-(C)/(2))=sin (C)/(2)\\cos C=1-2sin^2(C)/(2)\\(1)=2 sin (C)/(2) cos (A-B)/(2)-1+2sin^2\frac{C}2}\\=2sin(C)/(2)[cos (A-B)/(2)+sin (C)/(2)]-1~~~...(2)\\\\now~again~A+B+C=180\\C=180-(A+B)\\sin(C)/(2)=sin(90-(A+B)/(2))=cos (A+B)/(2)\\(2)=2sin\frac {C}{2}[cos (A-B)/(2)+cos (A+B)/(2)]-1\\`

`=-1+2sin(C)/(2)*2cos ((A-B)/(2) +(A+B)/(2) )/(2) cos ((A-B)/(2) -(A+B)/(2) )/(2) \\=-1+4sin(C)/(2) cos (A)/(2) cos(-B)/(2) \\=-1+4 cos (A)/(2) cos (B)/(2) sin (C)/(2)\\(cos(-B)=cos B)`