A disequation like this
-1 <= sinx <= 1
of course is always true, and is equivalent to
-1 <= sinx
sinx <= 1
What if you apply f(x) = x^-1 to both sides? It's a monotically decreasing function for x != 0, so you should just
reverse the disequality... but!
-1 >= 1/sinx
1 <= 1/sinx
which is nonsense. Am I wrong somewhere?