Derive a power reduction formula

Hi all... I've been working on a problem for the past several hours and haven't had much luck solving it...

I'm trying to derive this formula (for n>2):

And the hint that came with the challenge pointed out this trig identity:

I've played around with it a bit, but where I'm running into trouble is my u and dv values for integration by parts. I am fairly confident that I need to set my original integral like this:

That'll give me the I know I need, but no matter how I try, I'm not getting to the end result. I've played around with the identity, but it doesn't seem to be doing me a lot of good.

Any thoughts/solutions (in a hidden spoiler would be preferred, but at this point I'll take what I can get!)?

Thanks all...