Show that $\sin^220^\circ\sin40^\circ = \sin10^\circ \sin30^\circ \sin60^\circ$. This is the final step of one old competition problem, and in textbook it simply says "reader can prove this", but I found this a bit nontrivial. 

Show that  $\sin^220^\circ\sin40^\circ = \sin10^\circ \sin30^\circ \sin60^\circ$.

I tried a lot of things and formula on this, but never succeeded.
Can someone help? 
 A: Trigonometric Proof
This follows almost immediately from $$\sin 20^\circ \sin 40^\circ \sin 80^\circ =\frac{\sqrt 3}{8},\tag{1}$$
which is a corollary of
$$\sin x \sin (60^\circ-x) \sin (60^\circ+x)=\frac{\sin(3x)}{4}.$$
My hint is to write one of $\sin 20^\circ$ from $\sin^2 20^\circ$ as $2\sin 10^\circ \cos 10^\circ$, and to note that $\cos 10^\circ=\sin 80^\circ$.
Postscript:  We also have $$\cos x\cos(60^\circ-x)\cos(60^\circ+x)=\frac{\cos(3x)}{4},$$
making
$$\cos 20^\circ\cos 40^\circ \cos 80^\circ= \frac18.$$  Consequently
$$\tan x \tan(60^\circ -x )\tan(60^\circ+x)=\tan (3x)$$
and
$$\tan 20^\circ \tan 40^\circ \tan 80^\circ =\sqrt3.$$
We also have the following identity.
$$\prod_{j=1}^{n}\sin\left(x+\frac{(j-1)\pi}{n}\right)=\frac{\sin (nx)}{2^{n-1}}$$
for all positive integers $n$.  From this identity, 
$$\prod_{j=1}^{n}\cos\left(x+\frac{(j-1)\pi}{n}\right)=(-1)^{\frac{n-1}{2}}\frac{\cos (nx)}{2^{n-1}}$$
and
$$\prod_{j=1}^n\tan\left(x+\frac{(j-1)\pi}{n}\right)=(-1)^{\frac{n-1}{2}}\tan(nx)$$
for all odd positive integers $n$.  We also have
$$\prod_{j=1}^n\tan\left(x+\frac{(j-1)\pi}{n}\right)=(-1)^{\frac{n}{2}}$$
if $n$ is an even positive integer, which implies
$$\prod_{j=1}^{n}\cos\left(x+\frac{(j-1)\pi}{n}\right)=(-1)^{\frac{n}{2}}\frac{\sin (nx)}{2^{n-1}}$$
for each even positive integer $n$.

Geometric Proof

There is also a geometric proof without using $(1)$.  Let $ABC$ be a triangle with $\angle BAC=90^\circ$ and $\angle ABC=20^\circ$.  Let $E$ and $F$ be internal points of $AC$ and $AB$ such that $BE$ bisects $\angle ABC$ and $\angle ACF=30^\circ$.  We want to show that $\angle CFE=20^\circ$.
Note that $\angle BCF=40^\circ$.  If $D$ is an internal point of $BF$ such that $CD$ bisects $\angle BDC$, then $\triangle DBC$ is isosceles.  Therefore, $DM\perp BC$ if $M$ is the midpoint of $BC$. 
By the angular bisector theorem on $\triangle ABC$, $\frac{AE}{CE}=\frac{AB}{BC}$.  Since $\triangle MBD\sim \triangle ABC$, we have
$$\frac{DM}{BM}=\frac{BC}{AB}=\frac{CE}{AE}.$$
Since $BM=\frac12 BC$, we get
$$DB=\frac{CE}{AE} \cdot BM=\frac{BC\cdot CE}{2\cdot AE}.$$
Using the angular bisector theorem with $\triangle FCB$, we get $\frac{DF}{DB}=\frac{CF}{BC}$ so that
$$DF=\frac{CF}{BC}\cdot DB=\frac{CE}{AE}\left(\frac{CF}{2}\right).$$
Since $\triangle ACF$ is a right triangle with $\angle ACF=30^\circ$, we get $AF=\frac{CF}{2}$ so that $DF=\frac{CE}{AE}\cdot AF$ or
$$\frac{DF}{AF}=\frac{CE}{AE}.$$
This means $EF\parallel CD$.  Hence $$\angle CFE=\angle FCD=\frac{1}{2}\angle BCF=20^\circ.$$
From this result, we see using the law of sines on $\triangle CFE$ that
$$\frac{CE}{\sin 20^\circ}=\frac{CF}{\sin 130^\circ}=\frac{CF}{\sin 50^\circ}=\frac{CF}{\cos 40^\circ}.$$
Also the law of sines on $\triangle BCF$ yields
$$\frac{CF}{\sin 20^\circ}=\frac{BC}{\sin 120^\circ}=\frac{BC}{\sin 60^\circ}.$$
Therefore
$$\frac{CE}{\sin 20^\circ}=\frac{BC\sin 20^\circ}{\sin 60^\circ \cos 40^\circ}.\tag{2}$$
But using the law of sines with $\triangle BCE$ gives us
$$\frac{CE}{\sin 10^\circ}=\frac{BC}{\sin 100^\circ}=\frac{BC}{\sin 80^\circ}=\frac{BC}{2\sin 40^\circ \cos40^\circ}.\tag{3}$$
From $(2)$ and $(3)$ we obtain
$$\frac{BC \sin 10^\circ}{2\sin 40^\circ \cos 40^\circ}= CE=\frac{BC\sin^2 20^\circ}{\sin 60^\circ \cos 40^\circ}.$$
This proves that
$$\sin^2 20^\circ \sin 40^\circ =\frac{1}{2}\sin 10^\circ \sin 60^\circ =\sin 10^\circ \sin 30^\circ \sin 60^\circ.$$
And from this result, we can prove $(1)$ as well.
A: $LHS=\sin^2 20^\circ \sin 40^\circ =\sin 20^\circ\cdot\frac{1}{2}(\cos20^\circ -\cos60^\circ)$
$=\frac{1}{2}(\sin 20^\circ \cos20^ \circ-\frac{1}{2}\sin 20^\circ)=\frac{1}{4}(\sin 40^\circ-\sin 20^\circ)$
$ =\frac{1}{4}\cdot2\sin 10^\circ\cos 30^\circ=\frac14\sqrt3\sin 10^\circ. $
$RHS=\sin10^\circ \sin30^\circ \sin60^\circ=\frac14\sqrt3\sin 10^\circ.$
$LHS=RHS.$
A: $$F=4\sin20^2\circ\sin40^\circ$$
$$=2(1-\cos40^\circ)\sin40^\circ$$
$$2\sin40^\circ-\sin80^\circ$$
$$=\sin40^\circ-(\sin80-\sin40)$$ using 
http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html
$$F=\sin40-\sin20=?$$ 
