(b) If `(a+b+c)=0` then prove that, `x^(b^-1c^-1). x^(c^-1a^-1).x^(a^-1b^-1) =1`

Prove that: 1/(1+x^(a-b))+1/(1+x^(b-a))=1

If one of the roots of the equation ax^2 + bx + c = 0 be reciprocal of one of the a_1 x^2 + b_1 x + c_1 = 0 , then prove that (a a_1-c c_1)^2 =(bc_1-ab_1) (b_1c-a_1 b) .

If a+b+c=1, then prove that 8/(27a b c)>{1/a-1}{1/b-1}{1/c-1}> 8.

Prove that: (a+b+c)/(a^(-1)\ b^(-1)+b^(-1)\ c^(-1)+c^(-1)a^(-1))=a b c

If a > b > c >0 , prove that cot^(-1)((a b+1)/(a-b))+cot^(-1)((b c+1)/(b-c))+cot^(-1)((c a+1)/(c-a))=pi

If a,b,c are in A.P. prove that: a(1/b+1/c),b(1/c+1/a),c(1/a+1/b) are in A.P.

If 1/a ,1/b ,1/c are in A.P. prove that: (b+c)/a ,(c+a)/b ,(a+b)/c are in A.P.

If b is the harmonic mean between a and c, then prove that (1)/(b - a) + (1)/(b - c) = (1)/(a) + (1)/(c)

If a b c = 0, then ({(x^a)^b}^c)/({(x^b)^c}^a) = (a)3 (b) 0 (c) -1 (d) 1

If the lines a x+y+1=0,x+b y+1=0 and x+y+c=0(a ,b ,c being distinct and different from 1) are concurrent, then prove that 1/(1-a)+1/(1-b)+1/(1-c)=1.