form4 amaths

1 sinA+sinB+sin(A+B) = 4 cosA/2 cosB/2 sin(A+B)/2
SOLUTION
RHS
=4cos(A/2)cos(B/2)sin[(A+B)/2]
=4cos(A/2)cos(B/2)[sin(A/2)cos(B/2)+cos (A/2)sin(B/2)]
=4sin(A/2)cos(A/2)[cos(B/2)]^2+4sin(B/2)cos(B/2)[cos(A/2]^2
=2sinA[cos(B/2)]^2+2sinB[cos(A/2)]^2
=2sinA[1-{sin(B/2)}^2]+2sinB[1-{sin(A/2)}^2]
=2sinA-2sinA*[1/2(1-cosB)]+2sinB-2sinB*[1/2(1-cosA)]
=2sinA-sinA+sinAcosB +2sinB-sinB+sinBcosA
=sinA+sinB+sinAcosB+ cosAsinB
=sinA+sinB+sin(A+B)
=LHS
2
cos(A+B+C)+cos(-A+B+ C)+cos(A-B+C)+cos(A+ B-C) = 4cosAcosBcosC
SOLUTION
formula: cosAcosB =1/2[cos(A − B) + cos(A + B)]
cos(A+B+C) + cos(-A+B+C) + cos (A-B+C) + cos(A+B-C)
=cos(A+B+C) +cos(A+B-C)+ cos(A-B+C)+cos (A-B-C) [cos(-x)=cosx]
=2cos(A+B)cosC+2cos( A-B)cosC
=2cosC{cos[A+B]+cos[A-B]}
=4cosAcosBcosC