Identity involving an improper integral (with geometric application)

Since the main contribution to the integral comes from $t<<1$, analytically one has \begin{align} \lim_{c\to 0^+}\int_c^{\pi/2}\frac{c}{t}\sqrt\frac{1+t^2}{t^2-c^2}dt&=\lim_{c\to 0^+}\int_c^{\pi/2}\frac{c}{t}\sqrt\frac{1}{t^2-c^2}dt\\ &=\lim_{c\to 0^+}\int_{2/\pi}^{1/c}c\sqrt\frac{1}{1-c^2t^2}dt\\ &=\lim_{c\to 0^+}\int_{2c/\pi}^{1}\sqrt\frac{1}{1-t^2}dt\\ &=\int_{0}^{1}\sqrt\frac{1}{1-t^2}dt\\ &=\frac\pi{2} \end{align}

Your guess is correct. Let's write $t=cs$, so the integral becomes $$ \int_1^{\pi/(2c)} \sqrt{\frac{1+c^2 s^2}{s^2-1}}\, \frac{ds}{s} . $$ Fix a small $\epsilon>0$, and consider first the integral from $1$ to $\epsilon/c$. On this interval, if we drop the $c^2s^2$, we'll be off by not more than a multiplicative correction of order $(1+\epsilon)$, and then we're left with $$ \int_1^{\epsilon/c} \frac{ds}{s\sqrt{s^2-1}} = - \arctan\frac{1}{\sqrt{s^2-1}}\Bigr|_1^{\epsilon/c}\to \frac{\pi}{2} $$ as $c\to 0+$.

The remaining part of the integral, from $\epsilon/c$ to $\pi/(2c)$ is easily seen to be $O(c\log c^{-1})$, so this will go to zero when $c\to 0+$. Finally, we send $\epsilon\to 0$ to obtain the claim.

You can, if you want, calculate the integral. By performing the change of variables $$ t\mapsto \sqrt{\frac{1+t^2}{t^2-c^2}}. $$ you end up with a standard integral. For $0<c<\pi/2$ the result is $$ \int_c^{\pi/2}\frac{c}{t}\sqrt{\frac{1+t^2}{t^2-c^2}}\,dt=\lim_{a\to c^+}\biggl[c\,\text{artanh}\,\sqrt{\frac{t^2-c^2}{1+t^2}}-\arctan\biggl(c\sqrt{\frac{1+t^2}{t^2-c^2}}\biggr)\biggl]_a^{\pi/2}. $$ The first term will not contribute in the limit $c\to 0^+$, the second one will give you $\pi/2$.