If `a+b=1,a >0,` prove that `(a+1/a)^2+(b+1/b)^2geq(25)/2dot`

If a + b =1, a gt 0,b gt 0, prove that (a + (1)/(a))^(2) + (b + (1)/(b))^(2) ge (25)/(2)

If a+b=1, and a,b>0, then prove that (a+(1)/(a))^(2)+(b+(1)/(b))^(2)>=(25)/(2)

If a,b>0 prove that [(1+a)(1+b))^(3)>3^(3)a^(2)b^(2)

If a, b, c are in GP, prove that 1/((a+b)), 1/(2b), 1/(b+c) are in AP.

If a^(2)+b^(2)=1 , prove that (1+b+ia)/(1+b-ia)=b+ia .

If a,b,c are in G.P. then prove that (1)/(a+b),(1)/(2b),(1)/(b+c) are also in A.P.

If a, b, c are in G.P., then prove that (1)/(a^(2)-b^(2))-(1)/(b^(2)-c^(2))=-(1)/(b^(2)) . [Hint : Put b = ar, c = ar^(2) ]

Prove that: |1a a^2-b c1bb^2-c a1cc^2-a b|=0

In any DeltaABC , prove that (cos2A)/a^(2)-(cos2B)/b^(2)=(1/a^(2)-1/b^(2))

If a>0,b>0 and c>0 prove that (1984,2M(a+b+c)((1)/(a)+(1)/(b)+(1)/(c))>=9