STEP II 2007 question 5 solution

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(i)

(or induction if you're bored)

as 3 divides 2007.

(ii)

tan is periodic: since , it suffices to prove the result for n = 0, 1, 2 (and then, again, be as fussy as you like about the induction, which I won't).

as required.

as required.

as required.

(Cue tedious inductive step, or rather obvious conclusion as and )

(iii)

Bit of guesswork involved, but several hints give it away; |t| [U]<[/U] 1, the use of tan in part (ii), and the form of the expression suggest an expression in sin or cos. I chose sin:

Let so that

Note also that

Then the expression for g can be rewritten as:

And then another small inductive step... let so that Then:

etc.

Hence

Solution by generalebriety.