Simplifying $\frac{1-4\cos80^\circ}{\tan20^\circ}$ I am working on simplifying the value
$$\frac{1-4\cos80^\circ}{\tan20^\circ}$$
I am looking for it to be expressed in terms of tangent, because the value seems to be $\tan40^\circ$. However, I can't find any obvious way to do so. How would  simplify it to express like such?

Let $\theta = \pi/9 = 20^\circ$.  We wish to show that $$1 - 4 \cos 4\theta = \tan \theta \tan 2\theta. \tag{1}$$  This suggests the tangent angle addition identity $$\tan (\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \tag{2}$$ for the choice $$\alpha = \theta, \beta = 2\theta$$ which yields
$$(\tan 3\theta)(1 - \tan \theta \tan 2\theta) = \tan \theta + \tan 2\theta. \tag{3}$$  Since $3\theta = \pi/3 = 60^\circ$ hence $\tan 3\theta = \sqrt{3}$, we find
$$\begin{align}
1 - \tan \theta \tan 2\theta 
&= \frac{1}{\sqrt{3}}(\tan \theta + \tan 2\theta) \\
&= \frac{\sin \theta \cos 2\theta + \sin 2\theta \cos \theta}{\sqrt{3} \cos \theta \cos 2\theta} \\
&= \frac{\sin 3\theta}{\sqrt{3} \cos \theta \cos 2\theta} \\
&= \frac{1}{2\cos \theta \cos 2\theta}. \tag{4}
\end{align}$$
We now claim that $$8 \cos \theta \cos 2\theta \cos 4\theta = 1. \tag{5}$$  To this end, write
$$\begin{align}
8 \sin \theta \cos \theta \cos 2\theta \cos 4 \theta
&= 4 \sin 2\theta \cos 2\theta \cos 4\theta \\
&= 2 \sin 4\theta \cos 4\theta \\
&= \sin 8\theta \\
&= \sin(\pi - \theta) \\
&= \sin \theta,
\end{align}$$
and now dividing both sides by $\sin \theta$ yields the proof of $(5)$.  Therefore, $(4)$ is equivalent to
$$1 - \tan \theta \tan 2\theta = 4 \cos 4\theta,$$ which in turn is equivalent to $(1)$.

Something unsettled me about my solution above; I had the feeling it was unnecessarily complicated.  After some thought I noticed that it could be simplified as follows:  to show $(1)$, consider
$$\begin{align}
1 - \tan \theta \tan 2\theta 
&= \frac{\cos \theta \cos 2\theta - \sin \theta \sin 2\theta}{\cos \theta \cos 2\theta} \\
&= \frac{\cos 3\theta}{\cos \theta \cos 2\theta} \\
&= \frac{1}{2 \cos \theta \cos 2\theta},
\end{align}$$
which takes us directly to $(4)$ without the tangent addition identity.
