Confused about $\csc(x)$

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The title says it all, let's take this simple example:

I want to find $c$.

Using $\sin$ I get:

1. $\sin(30)$ = $4 / c$
2. $c = 4 / \sin(30) = 8$

Now, using $\csc$:

1. $\csc(30) = c / 4$
2. $c = \csc(30) * 4 = 8$

But also:

1. $\csc(30) = c / 4$
2. $\tan(30) = 4 / a$
3. $4 = \tan(30) * a$
4. $c = \csc(30)\cdot \tan(30)\cdot c = c\cdot (c / 4)\cdot (4 / a) = c$
5. OR: $\csc(30) \cdot \tan(30)\cdot c = \sec(30)\cdot c \approx c$

What's gone wrong here, I assumed $c$ would cancel out?

EDIT 1: Why isn't $c$ cancelling out in the last example, even after the fix?

EDIT 2: Fixed steps 2 and 3 (changed $c$ to $a$). Thank you for your help, silly mistakes.

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2 Answers

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$$\tan(30°)=\frac{4}{a}\text{ which isn't } \frac{4}{c}$$

Now we have that $$c=\text{cosec}(30°)·4\;\text{ and } \;4=\tan(30)°·a$$Thus $$c=\text{cosec}(30°)·\tan(30°)·a=\frac{c}{4}·\frac{4}{a}·a\not= \frac{c}{4}·\frac{4}{a}·\color{red}{c}$$ as pointed out in the comments

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$$\sec30^{\circ}=\frac{1}{\cos30^{\circ}}=\frac{2}{\sqrt3}.$$

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