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Prove the following:
46. $\dfrac{\csc\theta}{\cot\theta}-\dfrac{\cot\theta}{\csc\theta}=\tan\theta\sin\theta$

I got as far as
Right Side: $\tan\theta\sin\theta$ to $\dfrac{\sin\theta}{\cos\theta}\dfrac{\sin\theta}{1}$ and then; $\dfrac{\sin^2\theta}{\cos\theta}$

Left Side: $$\begin{align*} \dfrac{\csc\theta}{\cot\theta}-\dfrac{\cot\theta}{\csc\theta} &= \dfrac{\frac{1}{\sin^2\theta}-{\frac{\cos^2\theta}{\sin^2\theta}}}{\frac{\cos\theta}{\sin\theta}-{\frac{1}{\sin^2\theta}}}\\ &= \dfrac{\frac{1-\cos^2\theta}{\sin^2\theta}}{\frac{\cos\theta}{\sin^2\theta}}\\ &= \dfrac{\frac{\sin^2\theta}{\sin^2\theta}}{\frac{\cos\theta}{\sin^2\theta}}\\ &= \frac{1}{\frac{\cos\theta}{\sin^2\theta}}\\ &= \frac{\sin^2\theta}{\cos\theta} \end{align*}$$ Thanks a lot!

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    What "cross-cancelling"? You are subtracting the fractions, not multiplying them.2012-07-15

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There is no "cross cancelling". You are subtracting the fractions, not multiplying them.

$$\begin{align*} \frac{\csc\theta}{\cot\theta} - \frac{\cot\theta}{\csc\theta} & = \frac{\csc^2\theta - \cot^2\theta}{\cot\theta\csc\theta}\\ &= \frac{\quad\frac{1}{\sin^2\theta} - \frac{\cos^2\theta}{\sin^2\theta}\quad}{\frac{\cos\theta}{\sin\theta}\frac{1}{\sin\theta}}\\ &= \frac{\quad\frac{1 - \cos^2\theta}{\sin^2\theta}\quad}{\frac{\cos\theta}{\sin^2\theta}}. \end{align*}$$ Can you take it from there?

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    Yes I just realized that. I'm new to LaTeX. And I was trying to figure out how to divide fractions.2012-07-15
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    @Austin: Use `\frac`, not `\dfrac`. If you rightclick on my formula, you can ask that it show you the LaTeX code I used to produce it.2012-07-15
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    Thank you so much. You have been helpful today. And I'm re-editing my post with what I feel is correct. Stay posted!2012-07-15
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    Is the right side of my equation good? Or is there anything you have to say about that so I can look for equality on both sides.2012-07-15
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    @Austin: Your computations with $\tan\theta\sin\theta$ are correct. So you want to take the left hand side and change everything into sines and cosines, like I did, and simplify until you have a single fraction (instead of a compound fraction which is what I have up to where I developed it). You should be able to get it to be exactly the same as what you got for the right hand side.2012-07-15
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    I got it, thanks! And can you also tell me how to increase font size for a part of the body. Because when you divide fractions, the text can be microscopic!2012-07-15
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    @AustinBroussard: You can ask that the text be written in displaymode, but you do **not** want to write a complex fraction with all fractions in displaymode. The result is unwieldly: $$\frac{\ \displaystyle\frac{a}{b}\ }{\ \displaystyle\frac{c}{d}\ }\quad\text{vs.}\quad\frac{\ \frac{a}{b}\ }{\frac{c}{d}}.$$If you try it with a multiline formula, unless you get the spacing very well, it is easy to lose track of what is the numerator and what is the denominator, and the vertical space taken over is too large.2012-07-15
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    I will be adding a problem to this post for your assistance2012-07-15
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    @Austin: If it's a new problem, please consider writing a new question. Otherwise, the answer will no longer fit.2012-07-15