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?