I need help proving the following identity.
$$\tan^210^\circ+\tan^250^\circ+\tan^270^\circ=9$$
I am not sure if it is even true.
I need help proving the following identity.
$$\tan^210^\circ+\tan^250^\circ+\tan^270^\circ=9$$
I am not sure if it is even true.
If $\tan3y=\tan30^\circ$
$3y=n180^\circ+30^\circ$ where $n$ is any integer
$y=60^\circ n+10^\circ$ where $n=0,1,2$
For $n=2,y=130^\circ,\tan130^\circ=\tan(180^\circ-50^\circ)=-\tan50^\circ$
Now $\tan3y=\dfrac{3\tan y-\tan^3y}{1-3\tan^2y}$
and consequently, $\dfrac{3\tan y-\tan^3y}{1-3\tan^2y}=\dfrac1{\sqrt3}$ as $\tan30^\circ=\dfrac1{\sqrt3}$
Rearrange to form a cubic equation in $\tan y$ where $y=60^\circ n+10^\circ$ where $n=0,1,2$
We need $\tan^210^\circ+\tan^250^\circ+\tan^270^\circ$
$=(\tan10^\circ)^2+(-\tan50^\circ)^2+(\tan70^\circ)^2$
$=[\tan10^\circ+(-\tan50)+\tan70^\circ]^2$ $-2[\tan10^\circ(-\tan50)+(-\tan50)\tan70^\circ+\tan70^\circ\tan10^\circ]$