Grade 11Trigonometry

Let a=cosA + cosB -cos(A+B) and b= 4sin A/2 × sinB/2 × cos (A+B/2).then prove that a-b=1

Jaideep

7 Years agoGrade 11

1 Answer

Aditya Gupta

7 Years ago

define C= pi – (A+B)

then a – b= cosA + cosB + cosC – 4sin A/2 × sinB/2 ×sinC/2

now a= cosA + cosB + cosC = ( 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 )
= 1 + 4 sin(A/2) sin(B/2) sin(C/2)

= 1+ b

so that a – b=1

kindly approve :)