$\underset{\mathrm{t}\to 0}{lim} {\left(1^{\frac{1}{{\sin }^2\mathrm{t}}}+2^{\frac{1}{{\sin }^2\mathrm{t}}}+\dots .+{\mathrm{n}}^{\frac{1}{{\sin }^2\mathrm{t}}}\right)}^{{\sin }^2\mathrm{t}}$ is equal to 

$\begin{array}{l}\underset{t\to 0}{\lim }{\left({\mathrm{n}}^{\frac{1}{{\sin }^2\mathrm{t}}}\right)}^{{\sin }^2\mathrm{t}}{\left({\left(\frac{1}{\mathrm{n}}\right)}^{{\mathrm{cosec}}^2\mathrm{t}}+{\left(\frac{2}{\mathrm{n}}\right)}^{{\mathrm{cosec}}^2\mathrm{t}}+\dots \dots ..+1^{{\mathrm{cosec}}^2\mathrm{t}}\right)}^{{\sin }^2\mathrm{t}}\\ \left(\text{ As }\mathrm{n}\to \mathrm{\infty },{\mathrm{r}}^{\mathrm{n}}\to 0\text{ when }0<\mathrm{x}<1\right)\\ \left(\mathrm{n}\right)(0+0+\dots +1)=\mathrm{n}\end{array}$