Calculate a limit where $\sin n_k$ increases monotonically to 1. Let us consider numbers $n_k$ ($k=0,1,2,3,...$), where $\sin n_k$ increases monotonically to 1. Calculate
$$\lim_{k\rightarrow \infty} n_k\left(1-\sin n_{k}\right)-n_{k+1}\left(1-\sin n_{k+1}\right)$$
I tried to use this information:
$$\sin n_k\rightarrow 1\ \ \ \  (k\rightarrow\infty)$$
$$\sin n_k<\sin n_{k+1}\ \ \ \ \ \ \ \ \ \ $$
But without success. For example, what is the result of the limit
$$\lim_{k\rightarrow \infty} n_k\left(1-\sin n_{k}\right)$$
?
EDIT:
N.B.:this limit is obtained at the end of a miscalculation contained in a my research. However, this limit is very interesting and I'm trying to solve it!

Surly this depends on the $n_k$ you choose, if the choice is arbitrary, the limit can be every real number you like, because if you would like to obtain $x$. Just consider $x=\lim_{k\to\infty}2\pi q_k$, where $q_k=a_k/b_k$ is a rational number, $a_k,b_k$ are integers approaching infinity. Then we may choose $n_k = 2\pi a_k+ \arcsin{(1-\frac{1}{b_k})}.$ Then $\sin n_k = 1-\frac{1}{b_k}\to 1$. Also, 
$$\lim_{k\to\infty} n_k (1-\sin_{n_k})=\lim_{k\to\infty}2\pi \frac{a_k}{b_k} + o(1) = x.$$
