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My actual question doesn't have to do with what's said in the title, I'm having trouble with the derivative portion.

This is from the solution:

enter image description here

Source: cramster.com

I am fine from the point I have to take the derivative of "r" and all the plugging in that goes on, but the step right after that is what I'm confused about. I can't figure out how they came up with the final step you see above in the picture. What simplifying was done to obtain that part?

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Small typo - in the last denominator there's a $\sin^2$ that should be $\sin^2\theta$. –  Gerry Myerson Jul 17 '11 at 5:38
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up vote 1 down vote accepted

It doesn't seem complicated to me : $$ \cos \theta \sin \theta + (2+\sin \theta) \cos \theta = \cos \theta ( \sin \theta + 2 + \sin \theta) = \cos \theta (2 + 2 \sin \theta) = 2 \cos \theta (1 + \sin \theta) $$ for the numerator, and $$ \begin{align} \cos \theta \cos \theta - (2+ \sin \theta) \sin \theta & = \cos^2 \theta - 2 \sin \theta - \sin^2 \theta \\ & = 1 - \sin^2 \theta - 2 \sin \theta - \sin^2 \theta = 1 - 2 \sin \theta - 2 \sin^2 \theta. \end{align} $$

Sometimes it seems there is magic when there is not. It happens.

Hope that helps,

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