(0° ≤ α ≤ 90°, 0 ≤ α ≤ π/2) cot a=6

Cot a =6

So  adj=6.   Opp=1.     Hyp=sqrt(37)

You can take it from there :)

First find the angle a by using cot^-1(6)  (cot^-1 is also called acot or arccot).

$${acot}{\left({\mathtt{6}}\right)} = {\mathtt{9.462\: \!322\: \!208\: \!026^{\circ}}}$$  This is within the specified range 

Now 

$${csc}{\left({\mathtt{9.462\: \!322\: \!208\: \!026}}\right)} = {\mathtt{6.082\: \!762\: \!530\: \!311\: \!439\: \!4}}$$

You can try cos a, using any calculator with cos on it.

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any way you could put that in a fraction?

This picture might help:

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6/√37 can also be written as 6√37/37

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the problem is, i dont understand how you guys even got your answers. this stuff makes absolutely no sense to me

You need to memorise the expressions for sin, cos and tan in a right-angled triangle:

sin = opposite/hypotenuse

cos = adjacent/hypotenuse

tan = opposite/adjacent

and also memorise 

cot = 1/tan

csc = 1/sin

sec = 1/cos

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