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I have an equation $$x = \csc(\theta) - \cot(\theta).$$

As $\theta$ approaches zero, $x$ approaches zero. However, trying to solve the equation at zero yields an undefined result.

How do I rewrite the equation to be continuous at 0?

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Hint: Write the cosecant and cotangent in terms of sine and cosine. You can then combine the two fractions to give an expression that goes to 0/0. Expanding in a Taylor series or L'Hopital's rule will then be your friend.

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    Instead of Taylor series or l'Hôpital's rule, one can use the fact that $1 - \cos\theta = 2\sin^2(\theta/2)$, and then expand $\sin\theta$ in terms of $\theta/2$ as well.2011-01-18
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    Indeed, you can only treat this properly in terms of limits.2011-01-18
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    @Rahul Narain ah! thats it. So rewriting and canceling gives me $\tan(\theta/2)$!2011-01-18