Prove `(i)sinC+sinD=2sin(C+D)/2cos(C-D)/2 (ii) sinC-sinD=2sin(C-D)/2cos(C+D)/2`

Prove (i)sin C+sin D=2(sin(C+D))/(2)(cos(C-D))/(2)(ii)sin C-sin D=2(sin(C-D))/(2)(cos(C+D))/(2)

Prove (i)cos C+cos D=2(cos(C+D))/(2)(cos(C-D))/(2) (ii)cosC-cosD =2sin(C+D)/2sin(D-C)/2

Assertion (A) : Suppose that alpha- beta is not an odd multiple of (pi)/(2) , m in R-{0,1) and (sin (alpha+beta))/(cos (alpha-beta))=(1-m)/(1+m) then Tan ((pi)/(4)-alpha)=m cot ((pi)/(4)- beta) Reason (R) : for all C,D, & R cos c+cos D= 2 cos ((C+D)/(2)) cos((C-D)/(2)) cos-cosD=-2sin ((C+d)/(2))sin ((C-D)/(2))

Prove that : sinC +sinD =2 sin frac (C+D)(2) cos frac (C-D) (2) .

Prove that: sin(B-C)cos(A-D)+sin(C-A)cos(B-D)+sin(A-B)cos(C-D)

Prove that: sin(B-C)cos(A-D) + sin(C-A) cos (B-D) + sin(A-B) cos(C-D) = 0

cos(A-D)sin(B-C)+cos(B-D)sin(C-A)+cos(C-D)sin(A-B)=0

Show that: sin(B-C)cos(A-D)+sin(C-A)cos(B-D)+sin(A-B)cos(C-D)

In DeltaABC , prove that: a) (sin2A + sin2B + sin2C)/(sinA+sinB+sinC) = 8sinA/2 sinB/2sinC/2

Prove that : sinC-sinD = 2 cos frac (C+D)(2) sin frac (C-D) (2) .