How the property AM>or = GM used to get minimum value of the function......e,g for what condition of a and b does minimum value of a tan^2 x + b cot^2 x equals maximum value of a sin^2 x + b cos^2x

Ans) for min value of first func. we have..AM>or=GM

i.e      a.tan^2x  + b.cot^2x >or = 2(a.tan2x.bcot2x)^1/2    .....[ a+b/2 .or= (ab)^1/2 ]
                                                   = 2(ab)^1/2
hence min value of first function is 2(ab)^1/2

for second func.
its in the form of a.siny + b.cosy whose max value =(a^2 + b^2)^1/2
applying this we get the max. value of the func. as= (a^2 + b^2)^1/2

hence req. condition=> 2(ab)^1/2=(a^2 + b^2)^1/2
                                => 4ab = a^2 + b^2
                                =>  a^2 + b^2 -  4ab=0  (ANS)