Question 80871: Write
f(x) = cot (arccos (x/16))
in a way which doesn't involve trigonometric or inverse trigonometric functions
Answer by kev82(151)   (Show Source):
You can put this solution on YOUR website! Hi,
I'll start you off and hopefully you can finish it yourself. Knowledge of trig identities will come in handy here, whenever I encounter things like this, I find it easier to write the problem in terms of sin and cos, so let's do that. You may know that cot(a)=cos(a)/sin(a), but if you didn't, the way I remember it is that cot(a)=1/tan(a) and that tan(a)=sin(a)/cos(a). However you remember it doesn't matter, but knowing your cot(), cosec(), sec() in terms of sin() and cos() is very important.
Anyway, back to the problem, I'm going to let  as otherwise we're bound to leave a 16 out somewhere and get in a mess. So we're now looking at:
)}{\sin(\cos^{-1}(y))})
The top line is  right? Can you tell me why?
The bottom line needs a little more work. The way to get rid of the  is to take the cosine of it, like we did for the numerator. So we need a way of converting a sine to a cosine. There's a pretty good identity for that, I'm sure you've seen it:
Rearrange that, to get sin as a function of cosine, then substitute into the bottom line.
Once you've done that don't forget to rewrite it in terms of  , and that's your answer.
Once you've got it, or if you get stuck, post your answer/working and I can check/try to help.
Kev
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