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If a + b + c = 180 degree prove that cos square A + cos square B + cos square C + 2 cos A ×cosB× cosC= 1

If a + b + c = 180 degree prove that cos square A + cos square B + cos square C + 2 cos A ×cosB× cosC= 1

Grade:11

1 Answers

Daisy Srimayi
38 Points
6 years ago
Given A+B+C= 180
Taking LHS,
                    Cos^2A+cos^2 B + cos^2 C+ 2cosA cosB cosC
               =(1+cos2a/2)+ (1+cos2b/2)+ (1+cos2c/2)+2cosa cosb cosc
               =3/2+1/2(cos2a + cos2b + cos2c ) + 2cosa cosb cosc ........(1)
Take, cos2a+cos2b+cos2c
                   =2 (cos 2a+2b\2)(cos2a-2b\2) + cos2c
=2cos( a+b) cos (a-b)+cos2c
=2cos(180-c) cos (a-b)+cos 2c
=-2cosc cos(a-b)+2cos^2C - 1
-1-2cosc[ cos( a-b)-cosc]
=-1-2cosc[cos(a-b)-cos (180-(a+b))]
=-1-2cosc[cos(a-b)+cos(a+b)]
=-1-2cosc(2cosa cosb)
=-1-4cosa cosb cosc
Put this value in eq..(1)
LHS ,3\2+1\2(-1-4cosa cosb cosc) + 2 cosa cosb cosc
        =3\2-1\2-2cosa cosb cosc + 2 cosa cosb cosc
      =3\2-1\2=2\2=1 ....RHS
              LHS=RHS

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