Question

If, A+B+C=π

(i) Prove that, SinA+SinB-SinC = 4SinA/2. SinB/2. CosC/2
(ii) Prove that, Cos^2A + Cos^2B-Cos^2C = 1-2SinA.SinB.Cos.C
(iii) Prove that, CosA+CosB-CosC = -1+4CosA/2. CosB/2.SinC/2

Answer 1

(i)
A+B+C = pi
B+C = pi-A
sin(B+C) = sin(pi-A)
sin(B)cos(C)+cos(B)sin(C) = sin(pi)cos(A)-cos(pi)sin(A)
sin(B)cos(C)+cos(B)sin(C) = sin(A)
sin(B)cos(C) - sin(A) = -cos(B)sin(C)
(sin(B)cos(C) - sin(A))^2 = (-cos(B)sin(C))^2
sin^2(B)cos^2(C) + sin^2(A) - 2sin(A)sin(B)cos(C) = cos^2(B)sin^2(C)
-2sin(A)sin(B)cos(C) = cos^2(B)sin^2(C) - sin^2(B)cos^2(C) - sin^2(A)

(ii)
cos^2(A) + cos^2(B) - cos^2(C) = 1-2*sin(A)*sin(B)*cos(C)
cos^2(A) + cos^2(B) - cos^2(C) = 1+cos^2(B)sin^2(C) - sin^2(B)cos^2(C) - sin^2(A)
cos^2(A) + cos^2(B) - cos^2(C) = 1+(cos^2(B)sin^2(C) - sin^2(B)cos^2(C)) - sin^2(A) cos^2(A) + cos^2(B) - cos^2(C) = 1+((cos(B)sin(C))^2 - (sin(B)cos(C))^2) - sin^2(A)
cos^2(A) + cos^2(B) - cos^2(C) = 1+(cos(B)sin(C)-sin(B)cos(C))(cos(B)sin(C)+sin(B)cos(C)) - sin^2(A)
cos^2(A) + cos^2(B) - cos^2(C) = 1+(sin(C)cos(B)-cos(C)sin(B))(sin(C)cos(B)+cos(C)sin(B)) - sin^2(A)
cos^2(A) + cos^2(B) - cos^2(C) = 1+sin(C-B)sin(C+B) - sin^2(A)
cos^2(A) + cos^2(B) - cos^2(C) = 1+(1/2)*(cos(2B)-cos(2C)) - sin^2(A)
cos^2(A) + cos^2(B) - cos^2(C) = 1+(1/2)*(2cos^2(B)-1-(2cos^2(C)-1)) - sin^2(A)
cos^2(A) + cos^2(B) - cos^2(C) = 1+(1/2)*(2cos^2(B)-1-2cos^2(C)+1) - sin^2(A)
cos^2(A) + cos^2(B) - cos^2(C) = 1+(1/2)*(2cos^2(B)-2cos^2(C)) - sin^2(A)
cos^2(A) + cos^2(B) - cos^2(C) = 1+cos^2(B)-cos^2(C) - sin^2(A)
cos^2(A) + cos^2(B) - cos^2(C) = 1 - sin^2(A) + cos^2(B) - cos^2(C)
cos^2(A) + cos^2(B) - cos^2(C) = cos^2(A) + cos^2(B) - cos^2(C)