Solving limits without using L'Hôpital's rule $$\lim_{x \to \frac{\pi}{2}}\frac{b(1-\sin x) }{(\pi-2x)^2}$$
I had been solving questions like these using L'Hôpital's rule since weeks. But today, a day before the examination, I got to know that its usage has been 'banned', since we were never officially taught L'Hôpital's rule. 
Now I am at a loss how to solve questions I breezed through previously. While it is not possible to learn in 18 hours methods that would cover all kinds of problems, I am hoping I can pick up enough of them to salvage the examination tomorrow.
It has been hinted to me that the above could be solved using trigonometric techniques, but I'm at a loss as to how. 

we get $$\frac{(1-\sin(x))(1+\sin(x))}{(1+\sin(x))(\pi-2x)^2}$$ with $t=\pi-2x$ we get $$\frac{(\sin(t/2))^2}{4(\frac{t}{2})^2}$$
