Question

show that ,cosA+cosB+cosC=1+4sinA/2×sinB/2×sinC/2

Answer 1

  A + B + C = π ...... (1) 
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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. ............................. Q.E.D.

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In step (2), we used the Factorization formula 

cos x - cos y = 2 sin [ (x+y)/2 ] · sin [ (y-x)/2 ] 

Answer 2

Answer:

use the factorization formula

may it will helpful to you