Does tan300-tan30=tan270?

Jun 25, 2018

NO. See explanation below

Explanation:

We know that $270 = 300 - 30$

In goniometric circle we see that 270 does not have tangent value. But let use tangent of diference to demonstrate analitically...

We know also that $\tan \left(x - y\right) = \frac{\tan x - \tan y}{1 + \tan x \tan y}$

Lets check the formula in our case

$\tan \left(270\right) = \tan \left(300 - 30\right) = \frac{\tan 300 - \tan 30}{1 + \tan 300 \tan 30} =$

And identity proposed will be thrue if $1 + \tan 300 \tan 30 = 1$

Lets see:
$\tan 300 = - \sqrt{3}$
$\tan 30 = \frac{\sqrt{3}}{3}$

tan300tan30=-sqrt3·sqrt3/3=-1 then denominator is $1 - 1 = 0$

That means $\tan 270$ does not exist, then identity proposed is not possible

Jun 25, 2018

$\tan {300}^{\circ} - \tan {30}^{\circ} \ne \tan {270}^{\circ}$.
$\tan {270}^{\circ}$ is undefined.
$\tan {300}^{\circ} - \tan {30}^{\circ} = - \frac{4}{\sqrt{3}} = - \frac{4}{3} \sqrt{3}$.

Explanation:

$\tan {270}^{\circ}$ is undefined.

In fact, $\tan {300}^{\circ} - \tan {30}^{\circ} = \tan \left({360}^{\circ} - {60}^{\circ}\right) - \frac{1}{\sqrt{3}}$,

$= - \tan {60}^{\circ} - \frac{1}{\sqrt{3}} = - \sqrt{3} - \frac{1}{\sqrt{3}} = - \frac{4}{\sqrt{3}} = - \frac{4}{3} \sqrt{3}$.