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Prove that (sec^4alpha)/(tan^2beta)+(sec...

Prove that `(sec^4alpha)/(tan^2beta)+(sec^4beta)/(tan^2alpha)ge8`. If each term in the expression is well defined.

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Given that a,b,c are positive real numbers. To prove that
`(a + 1)^(7) + (b + 1)^(7) (c + 1)^(7) gt 7^(7) a^(4) b^(4) c^(4)`.
`L.H.S = (1 + a)^(7) (1 + b)^(7) (1 + c)^(7)`
`= [(1 + a) (1 + b) (1 + c)]^(7)` (1)
Now, A.M. `ge` G.M
`implies (a + b + c + ab + bc + ca + abc)/(7) ge (a^(4) b^(4) c^(4))^(1//7)`
or `(a + b + c +ab + bc + ca + abc)^(7) ge 7^(7) (a^(4) b^(4) c^(4))` (2)
From Eqs. (1) and (2) we get
`[(1 + a)(1 + b)(1 + c)]^(7) gt 7^(7) a^(4)b^(4)c^(4)`
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