Induction of sequence

Question

I am really stuck on one of my excersies. I know it's true for $n=1$, but I can figure out how to prove it for $k+1$. This excersise it considerably more difficult than the one we discusses in class. I have only done induction in series. Is there any way to write $x_n$ so it makes more sense?

We have: $$x_n = \cos^2(x_{n-1})\sin(x_{n-2}).$$ Show by induction that $$0\le x_n \le 1$$ for all whole numbers $$n\ge 2$$

where: $$ x_0 = \pi/2,\quad x_1 = 3$$ $x_n$ is a sequence.

Answer 1

Hint: you can use a stronger form of induction, where the inductive hypothesis is extended to all terms preceding $x_k$. So you may assume that both $0\le x_{k-1}\le1$ and $0\le x_{k-2}\le1$.

Answer 2

Hint:

You known that $0\le\cos^2(x)\le1$ and $-1\le\sin(x)\le1$.

Then it suffices to prove that $\sin(x_{n-2})$ remains positive.