If $\mathrm{cosec}\frac{\mathrm{\pi }}{32}+\mathrm{cosec}\frac{\mathrm{\pi }}{16}+\mathrm{cosec}\frac{\mathrm{\pi }}{8}+\mathrm{cosec}\frac{\mathrm{\pi }}{4}+$$\mathrm{cosec}\frac{\mathrm{\pi }}{2}=\cot \frac{\mathrm{\pi }}{\mathrm{k}}$ then the value of k is

$\begin{array}{l}{\mathrm{T}}_1=\mathrm{cosec}\mathrm{\theta }=\frac{\sin \left(\mathrm{\theta }-\frac{\mathrm{\theta }}{2}\right)}{\sin \frac{\mathrm{\theta }}{2}\sin \mathrm{\theta }};\mathrm{\theta }=\frac{\mathrm{\pi }}{32}\\ {\mathrm{T}}_1=\cot \frac{\mathrm{\theta }}{2}-\cot \mathrm{\theta }\\ {\mathrm{T}}_2=\cot \mathrm{\theta }-\cot 2\mathrm{\theta }\\ {\mathrm{T}}_3=\cot 2\mathrm{\theta }-\cos 2^2\mathrm{\theta }\\ {\mathrm{T}}_4=\cot 2^2\mathrm{\theta }-\cos 2^3\mathrm{\theta }\\ {\mathrm{T}}_5=1\\ \text{ Sum }=1+\cot \frac{\mathrm{\theta }}{2}-\cot 8\mathrm{\theta }\\ =1+\cot \frac{\mathrm{\pi }}{64}-\cot \frac{\mathrm{\pi }}{4}=\cot \frac{\mathrm{\pi }}{64}=\cot \frac{\mathrm{\pi }}{\mathrm{k}}\therefore \mathrm{k}=64\end{array}$