a-maths~~~~~~~

a)L.H.S.=sin²(A/2)+sin²(B/2)+sin²(C/2)
=(1-cosA)/2+sin²(B/2)+(1-cosC)/2
=(2-cosA-cosC)/2+ sin²(B/2)
=[2-2(1-cosB)]/2+ sin²(B/2)
=1-1+cosB+1- cos²(B/2)
=2cos²(B/2)-1+1- cos²(B/2)
=cos²(B/2)
=R.H.S.
b)cosA+cosC=2(1-cosB)
2cos[(A+C)/2]cos[(A-C)/2]=2(1-cos[180*-(A+C)])
cos[(A+C)/2]cos[(A-C)/2]=1+cos(A+C)
cos[(A+C)/2]cos[(A-C)/2]=1+2cos2[(A+C)/2]-1
cos[(A+C)/2]cos[(A-C)/2]=2cos2[(A+C)/2]
cos(A-C)/2=2cos[(A+C)/2]
L.H.S.= tan(A/2)tan(C/2)
= sin(A/2)sin(C/2)/cos(A/2)cos(C/2)
= (1/2)[cos [(A-C)/2] – cos[(A+C)/2]/(1/2)[cos[(A+C)/2]+cos[(A-C)/2]]
= [2cos[(A+C)/2] – cos[(A+C)/2]]/[cos[(A+C)/2]+2cos[(A+C)/2]]
= 1/3
= R.H.S.
ci)L.H.S.=cot(A/2)+cot(C/2)
=[tan(C/2)+tan(A/2)]/tan(A/2)tan(C/2)
=[tan(C/2)+tan(A/2)]/(1/3)
=3[tan(C/2)+tan(A/2)]
=R.H.S.
ii)R.H.S.=cot(A/2)+cot(C/2)
=[tan(A/2)+tan(C/2)][1-tan(A/2)tan(C/2)]/[tan(A/2)tan(C/2)][1-tan(A/2)tan(C/2)]
=tan[(A+C)/2](1-1/3)/(1/3)
=2tan[(180*-B)/2]
=2tan(90*-B/2)
=2cot(B/2)
=L.H.S.