In A B C if x= ( B-C 2 ) A 2 , y= ( C-A 2 ) B 2 , and z= ( A-B 2 ) C 2 , then (x+y+z) is equal to

Step-by-step solution

In ABC , given aligned & x= ( B-C 2 ) A 2 & y= ( C-A 2 ) B 2 aligned and y= ( A-B 2 ) C 2 Since, ( B-C 2 )= b-c b+c A 2 x= b-c b+c Similarly, y= c-a c+c and z= a-b a+b Now, by componendo and dividendo 1+x 1-x = b+c+b-c b+c-b+c = b c Similarly, 1+y 1-y = c a and 1+z 1-z = a b aligned & ( 1+x 1-x ) ( 1+y 1-y ) ( 1+z 1-z )= b c c a a b =1 & (1+x)(1+y)(1+z) = & (1-x)(1-y)(1-z) & 1+x+y+z+x y+y z+z x+x y z = & 1-(x+y+z)+x y+y z+z x-x y z & x+y+z=-x y z aligned