sin/cos = cos/sin

Question

To begin with, let's note the following:

$$ \tan = \frac{\sin}{\cos} \\ \sin = \tan \cdot \cos \\ \cos = \frac{\sin}{\tan} $$

If we replace one of the functions in $\frac{\cos}{\sin}$ according to the definitions above:

$$ \frac{\cos}{\sin} = \frac{\frac{\sin}{\tan}}{\sin} = \frac{{\sin}}{\tan\sin} = \tan $$

or:

$$ \frac{\cos}{\sin} = \frac{\cos}{\tan\cos} = \tan $$

So it looks like $\sin\div\cos = \cos\div\sin$ what can't be truth (except for 45°×n). I must have made a mistake somewhere but I am not sure where. How come?

Answer 1

You cancelled improperly: $$\require{cancel} \frac{\sin \theta}{\tan \theta \sin \theta}= \frac{\cancel{\sin \theta}}{\tan \theta \cancel{\sin \theta}}= \frac{1}{\tan \theta }=\cot \theta $$

Answer 2

Note,

$$\frac{1}{\tan x}=\cot x.$$