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I appreciate the help.

My attempt:

$$ \begin{align} \tan + \cot &= \frac{\sin}{\cos} + \frac{\cos}{\sin} \\ &= \frac{\sin^2}{\cos \sin}+\frac{\cos^2}{\cos \sin} \\ &= \frac{\sin^2+\cos^2}{\cos \sin}\\ &= \frac{1}{\cos \sin}\\ &= \frac{1}{\frac{1}{\sec}\frac{1}{\csc}}\\ &=\frac{1}{\frac{1}{\sec \csc}}\\ &=\frac{1}{1}\cdot \frac{\sec \csc}{1}\\ &= \sec \csc \end{align} $$

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OK! If you have to do this for an exam, however, I suggest you write in all of the " $ \ \theta \ $ "s (or whatever symbol you are using for angles). A grader may take points off for not writing the functions properly. (What you did is fine for your own "scrap work", of course.) –  RecklessReckoner Feb 3 at 0:48
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yup. It's quicker to go from $\frac1{cos\cdot{sin}}$ to $\frac1{cos}\frac1{sin}=sec\cdot{csc}$. –  Eleven-Eleven Feb 3 at 0:49
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1 Answer 1

That is exactly correct! Just two things: First, $\tan,\sin,\cos,$ etc hold no meaning on their own, they need an argument. So just be sure to write $\tan x$, $\cos x$ etc rather than just $\tan$ or $\cos$.

Finally, you could save time on your proof by noticing on the fourth step that $$ \frac{1}{\cos x\sin x}=\frac{1}{\cos x}\frac{1}{\sin x}=\sec x \csc x $$

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