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Let alpha,beta,gamma>0a n dalpha+beta+ga...

Let `alpha,beta,gamma>0a n dalpha+beta+gamma=pi/2dot` Then prove that `sqrt(tanalphatanbeta)` + `sqrt(t a nbeta"tan"gamma)+sqrt(tanalphatangamma)lt=sqrt(3)`

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Let `p=tan alpha beta,q=tan beta gamma "and" r=tan alpha tan gamma`
Now, `(sqrt(p)+sqrt(q)+sqrt(4))^(2)=p+q+r+2sqrt(pq)+2sqrt(qr)_2sqrt(rp)`
Also, using `GM leAM` we have
`2sqrt(pq)lep+q,2sqrt(qr)leq+r,2sqrt(rp)ler+p`
`therefore (sqrt(p)+sqrt(1)+sqrt(r))^(2)lep+q+r+p+q+q+r+r+p`
`therefore sqrt(p)+sqrt(q)+sqrt(r)lesqrt(3).sqrt(p+q+r)`
Now for `alpha+beta+gamma=(pi)/(2)`, we have
`p+q+r=tan alpha tan beta+tan tan gamma+tan alpha tan gamma=1`
`therefore sqrt(p)+sqrt(q)+sqrt(r)ltsqrt(3)`
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