32 If A+B+C= 180°, then prove that cosAtosB+cosC=1+4sin2 sing sine

1.	32 If A+B+C= 180°, then prove that cosAtosB+cosC=1+4sin2 sing sine
Answer» A + B + C = π ...... (1)L.H.S. = ( cos A + cos B ) + cos C = { 2 · cos[ ( A+B) / 2 ] · cos [ ( A-B) / 2 ] } + cos C = { 2 · cos [ (π/2) - (C/2) ] · cos [ (A-B) / 2 ] } + cos C = { 2 · sin( C/2 ) · cos [ (A-B) / 2 ] } + { 1 - 2 · sin² ( C/2 ) } = 1 + 2 sin ( C/2 )· { cos [ (A -B) / 2 ] - sin ( C/2 ) } = 1 + 2 sin ( C/2 )· { cos [ (A-B) / 2 ] - sin [ (π/2) - ( (A+B)/2 ) ] } = 1 + 2 sin ( C/2 )· { cos [ (A-B) / 2 ] - cos [ (A+B)/ 2 ] } = 1 + 2 sin ( C/2 )· 2 sin ( A/2 )· sin( B/2 ) ... ... ... (2) = 1 + 4 sin(A/2) sin(B/2) sin(C/2) = R.H.S.

Answer»

A + B + C = π ...... (1)L.H.S.

= ( cos A + cos B ) + cos C

= { 2 · cos[ ( A+B) / 2 ] · cos [ ( A-B) / 2 ] } + cos C

= { 2 · cos [ (π/2) - (C/2) ] · cos [ (A-B) / 2 ] } + cos C

= { 2 · sin( C/2 ) · cos [ (A-B) / 2 ] } + { 1 - 2 · sin² ( C/2 ) }

= 1 + 2 sin ( C/2 )· { cos [ (A -B) / 2 ] - sin ( C/2 ) }

= 1 + 2 sin ( C/2 )· { cos [ (A-B) / 2 ] - sin [ (π/2) - ( (A+B)/2 ) ] }

= 1 + 2 sin ( C/2 )· { cos [ (A-B) / 2 ] - cos [ (A+B)/ 2 ] }

= 1 + 2 sin ( C/2 )· 2 sin ( A/2 )· sin( B/2 ) ... ... ... (2)

= 1 + 4 sin(A/2) sin(B/2) sin(C/2)

= R.H.S.

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