$\underset{n\to \infty }{\lim }{n}^2\sqrt{\left(1-\cos \frac{1}{n}\right)\sqrt{\left(1-\cos \frac{1}{n}\right)\sqrt{\left(1-\cos \frac{1}{n}\right)\sqrt{\mathrm{...}to\infty }}}}$ equals

$\underset{n\to \infty }{Lt}{n}^2\sqrt{\left(1-\cos \frac{1}{n}\right)\sqrt{\left(1-\cos \frac{1}{n}\right)\sqrt{\mathrm{.....}\infty }}}$

$y=\sqrt{\left(1-\cos \frac{1}{n}\right)\sqrt{\left(1-\cos \frac{1}{n}\right)\sqrt{\mathrm{.....}\infty }}}$

$y^2=\left(1-\cos \frac{1}{n}\right)y$$\Rightarrow y=1-\cos \frac{1}{n}$

$\underset{n\to \infty }{Lt}{n}^2\left(1-\cos \frac{1}{n}\right)=\underset{\frac{1}{n}\to 0}{Lt}\frac{1-\cos \frac{1}{n}}{{\left(\frac{1}{n}\right)}^2}=\frac{1}{2}$