Q:

How to differentiate cos(x) where x is in steradians?

Its been long time I did some differentiation, nevertheless I went through the basic differentiation of all trigonometric functions but couldn't understand why the differentiation of $cos(x)$ is $2 \pi sin(x)$, where $x$ is in Steradians. Please let me if I am missing something or let me know the right way to understand it. Many thanks.
Edit: Here is the expression I am talking about. It's actually a formula for solid angle. 
$d$ = $(d/^2) _0$  = $2  sin_1 d_1 _0$.
Here is the picture describing the parameters

A:

The formula you have is for the solid angle between $x$ and $x + \varepsilon_1$, where $\varepsilon_1$ is like $dx$.  The $2\pi $ comes from integrating around the azimuthal angle.

