Prove that `(a+b)^(-1)(a^(-1)+b^(-1))=(1)/(ab)`

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

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

Prove that, tan^(-1)a+tan^(-1)b+tan^(-1)((1-a-b-ab)/(1+a+b-ab))=(pi)/(4) .

Prove that tan^(-1)((a-b)/(1+ab))+ tan^(-1)((b-c)/(1+bc))+tan^(-1)((c-a)/(1+ca))=0 , ab>(-1), bc>(-1), ca>(-1)

Prove that (AB)^(-1) = B^(-1) . A^(-1) give that A = [(2,3),(1,-4)] and B = [(1,-2),(-1,3)] .

Prove that : tan^(-1) a - tan^(-1) b = cos ^(-1) [(1+ab)/(sqrt((1+a^(2))(1+b^(2))))]

Prove that : tan^(-1) a - tan^(-1) b = cos ^(-1) [(1+ab)/(sqrt((1+a^(2))(1+b^(2))))]

Prove that : cos ^(-1) ((1- a^(2))/(1+a)) + cos ^(-1)((1-b^(2))/(1+b^(2))) = 2 tan ^(-1) .(a+b)/(1-ab)

Prove that: cot^(-1)((ab+1)/(a-b))+cot^(-1)((bc+1)/(b-c))+cot^(-1)((ca+1)/(c-a))=0

Prove that: cot^(-1)((ab+1)/(a-b))+cot^(-1)((bc+1)/(b-c))+cot^(-1)((ca+1)/(c-a))=0 .