In my research work, I need to show that the sequence $(nu_n)$  tends to 0 where $ (u_n)$ is defined by  $$u_{n+1}=u_{n} \cos^{2}(n),\quad  u_{0}=1$$ 
$(u_n)$ is a positive and decreasing sequence. My adempt \begin{align*}
u_{n+1}&= \prod_{k=1}^n\cos^2(k)  =(\prod_{k=1}^n \frac{e^{k i } + e^{- k i }}{2} )^2 \\
 &= 4^{-n}e^{-2(1+2+\cdots+n) \cdot i } \prod_{k=1}^n \left( 1 + e^{2 k i } \right)^2  \\
&=4^{-n}e^{-n(n+1)\cdot i } \prod_{k=1}^n \left( 1 + e^{2 k i } \right)^2 
\end{align*}  Afterwards, I don't see how to continue