Prove that `(a b+x y)(a x+b y)>4a b x y(a , b ,x ,y >0)dot`

If x+y+z=0 prove that |a x b y c z c y a z b x b z c x a y|=x y z|a b cc a bb c a|

Prove that (b+x)(c+x),(v+x)(a+x),(a+x)(b+x)(b+y)(c+y),(c+x)(a+t),(a+y)(b+y)(b+z)(c+z),(c+z)(a+z),(a+z)(b+z)]|=(b-c)(c-a)(y-z)(x-y)

(i) If tanA-tan B=x and cot B- cot A=y . Prove that cot(A-B)=1/x + 1/y . (ii) If 2 cos A = x+1/x, 2 cos B = y + 1/y , then show that 2 cos (A-B)= x/y+y/x .

If x ,\ y , z are not all zero and if a x+b y+c z=0; b x+c y+a z=0; c x+a y+b z=0 Prove that x : y : z=1:1:1\ or\ 1:omega:omega^2or\ 1:omega^2:omegadot

Given that x , y a n d b are real numbers and x >0 then x/b y/b d. x/bgeqy/b