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I have recently acquired the fact that $\tan(x)$ is equal to... $$\frac{x}{1-\frac{x^2}{3-\frac{x^2}{5-\frac{x^2}{7-\frac{x^2}{9-...}}}}}$$ It is also known that $\tan(x)$ is equal to... $$\frac{\sin(x)}{\cos(x)}$$

Why does it not follow that $sin(x)$ is equal to... $$x$$ and $\cos(x)$ is equal to... $${1-\frac{x^2}{3-\frac{x^2}{5-\frac{x^2}{7-\frac{x^2}{9-...}}}}}$$ Both of the fractions $\frac{x}{1-\frac{x^2}{3-\frac{x^2}{5-\frac{x^2}{7-\frac{x^2}{9-...}}}}}$ and $\frac{\sin(x)}{\cos(x)}$ are simplified, therefore their numerators should be equivalent to each other, as well as their denominators.

In the first relation, x is the numerator and ${1-\frac{x^2}{3-\frac{x^2}{5-\frac{x^2}{7-\frac{x^2}{9-...}}}}}$ is the denominator. In the second relation, $\sin(x)$ is the numerator and $\cos(x)$ is the denominator, yet the result that logically makes sense (that the numerators would be equivalent and the denominators would be equivalent) is incorrect.

Apologies if I am missing something obvious.

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    $\begingroup$ What does it mean to say that a quotient of one real number by another is "simplified"? $\endgroup$ – lulu Jun 5 '16 at 19:41
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Well, you can say that $3/3=1/1$ but this does not imply $3=1$. I believe (in essence) this is the mistake you're making despite the "simplification" comment.

Update #1:

I think we need a "formal" definition of simplified here. Regardless, I admit that the $3/3$ example is not the best. I believe (as @Teepeemm commented) that "simplified" only makes sense for rational numbers (ratios of integers).

Update #2: If you want to extend the meaning of "simplified" to include the definition you gave:

A simplified fraction consists of a numerator and denominator that have a greatest factor of 1. The GCF of the numerator and denominator in the fractions expressed in the OP is 1. Every number is divisible by one, and irrational numbers pose no exceptions. Is there a more formal definition?

Then, if $a/b=3/7$ you can't say whether $a=3/7$ and $b=1$ or $a=3$ and $b=7$ (it ends up being ambiguous).

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  • $\begingroup$ As I said, both of these fractions are simplified (unless I am making an invalid assumption there). 3/3 is not simplified. $\endgroup$ – Mar Dev Jun 5 '16 at 19:38
  • $\begingroup$ @MarDev "Simplified" really only makes sense for a rational number, with an integer over another integer. That won't happen for both of your fractions at the same time. $\endgroup$ – Teepeemm Jun 5 '16 at 19:41
  • $\begingroup$ @Teepeemm 4(pi)/2(pi) simplifies to 2/pi, yet both the 4(pi) and 2(pi) are irrational. A simplified fraction consists of a numerator and denominator that have a greatest factor of 1. The GCF of the numerator and denominator in the fractions expressed in the OP is 1. Every number is divisible by one, and irrational numbers pose no exceptions. Is there a more formal definition? $\endgroup$ – Mar Dev Jun 5 '16 at 19:53
  • $\begingroup$ The last example is what made the answer complete. Didn't comprehend that case. Why did you edit it out? $\endgroup$ – Mar Dev Jun 5 '16 at 19:58
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    $\begingroup$ I think, formally, as @Teepeemm mentioned, "simplified" only makes sense for rational numbers so my latest update was incorrect given that definition. I'll add it back as per your latest comment! $\endgroup$ – karlahrnndz Jun 5 '16 at 20:03
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The coefficients are to be in proportion.

$$ \frac{a}{c}=\frac{b}{d}$$

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