Then this reduces to:
Applying the half-angle identity
now bring the
under the square root in the middle term.
This is obviously true if we can get that
since all terms are positive, squaring preserves ordering, so examine the resultant expression:
this is true iff
Now the result is obvious, because the
is an even function, so has all even terms in its taylor series, and the second order approximation is
, in particular it is evident that the inequality is true, hence--as each step was reversible, the argument shows the inequality to be true.