Prove that `[(a^2+b^2)/(a+b)]^(a+b)> a^a b^b >{(a+b)/2}^(a+b)dot`

Prove that 2/(b+c)+2/(c+a)+2/(a+b) 0.

Prove that b^2c^2+c^2a^2+a^2b^2> a b cxx(a+b+c)(a ,b ,c >0) .

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

Prove that (b^2+c^2)/(b+c)+(c^2+a^2)/(c+a)+(a^2+b^2)/(a+b)> a+b+c

In Delta ABC, prove that (a^(2) - b^(2) + c^(2)) tan B = (a^(2) + b^(2) - c^(2)) tan C.

Prove that matrix [((b^(2)-a^(2))/(a^(2)+b^(2)),(-2ab)/(a^(2)+b^(2))),((-2ab)/(a^(2)+b^(2)),(a^(2)-b^(2))/(a^(2)+b^(2)))] is orthogonal.

Prove that |a b+c a^2b c+a b^2c a+b c^2|=-(a+b+c)xx(a-b)(b-c)(c-a)dot

Prove that |a b+c a^2b c+a b^2c a+b c^2|=-(a+b+c)xx(a-b)(b-c)(c-a)dot

In a Delta ABC, prove that (a + b)/(a - b) = tan ((A + B)/(2)) cot ((A - B)/(2))

If sin^(-1)((2a)/(1+a^2))+sin^(-1)((2b)/(1+b^2))=2tan^(-1)x , then x is equal to [a , b , in (0,1)] (a) (a-b)/(1+a b) (b) b/(1+a b) (c) b/(1+a b) (d) (a+b)/(1-a b)