Finding range of $f(x) = \sin^4 x\tan x + \cos^4 x\cot x$ I got to a certain step and couldn't continue. I can't fully understand the provided solution...
$$ f(x) = {\sin^6x+\cos^6x \over \sin x \cos x} = {2-1.5\sin^2 2x \over \sin 2x}$$
Let$$ t=\sin2x, t \neq 0$$
$$ f(x) = g(t) = {2 \over t} - {3t \over 2}$$
I don't get the next part.
Since $$\frac2t\ \text{and}\ -\frac{3t}2$$
are both decreasing on $$[-1,0)\cup(0,1]$$
The range of $g(t)$ is $$(-\infty , -0.5]\cup[0.5,+\infty)$$ which is the range of $f(x)$.
Any explanation would be greatly appreciated.

The Intermediate Value Theorem is implicitly used. If a continuous function $f$ is decreasing on a interval $[a,b)$ the range of $f$ on this interval is 
$$\bigl(f(b_-), f(a)\bigr],\quad\text{with}\enspace f(b_-)=\lim_{\substack{x\to b\\x<b}}f(x).$$
Similarly for intervals of type $(a,b]$ or $(a,b)$. Exchange the bounds if the function is increasing.
