I don't quite understand what you mean by $x \to 45^\circ-x$, but what you wish to convey is rather intuitive. You'd like to say, that the terms will cancel away, which is easy to see as numerator and denominator are the sines of same arguments but written reverse. So, I hope this helps
EDIT: It's true that this is the age old of trick called Change of variable, but, I don't see a notation that incorporates the change in limit that occurs with the change in variable. This is just a matter of notation and people's way of doing math.
Your Idea is right and you are through.
ADDITION: (Incorporated from @Srivatsan's link to this question and his answer)
An alternative approach will be to observe the following:
I'll assume that you know that $\tan(x+y)=\dfrac{\tan x + \tan y}{1-\tan x \tan y}$ and point you to the following rather nice way of doing it:
$\cot(x+y)=\dfrac{1}{\tan(x+y)}=\dfrac{1-\tan x \tan y}{\tan x + \tan y}=\dfrac{\cot x \cot y-1}{\cot x + \cot y} $
Now, if $ x+y = 45^\circ$, then $\cot x \cot y-1=\cot x + \cot y$ Tweak this a little to observe, $ (1-\cot x)(1-\cot y)=2$
Now, see that your product coincides with $22$ such pairs $(1,44); (2, 43);....(22,23)$ and this gives you the answer.