Rat on. Yeah, so the trick is to see that perp. from abscissa up to the radius (i.e., from right angle up to hypotenuse) is the same as <a> times <b>, which IS the product of sine and cosine. So that's the geometric interpretation, pretty cool. For the derivative, just do the chain rule, like you said--so it's cos-squared minus sin-squared, which lurches all over the place as you rotate counterclockwise around the unit circle. Who'da thunk!
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