Question

Does `tan300-tan30=tan270`?

Answer 1

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(x-y)=(tanx-tany)/(1+tanxtany)`

Lets check the formula in our case

`tan(270)=tan(300-30)=(tan300-tan30)/(1+tan300tan30)=`

And identity proposed will be thrue if `1+tan300tan30=1`

Lets see:
`tan300=-sqrt3`
`tan30=sqrt3/3`

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

That means `tan270` does not exist, then identity proposed is not possible

Answer 2

`tan300^@-tan30^@!=tan270^@`.
`tan270^@` is undefined.
`tan300^@-tan30^@=-4/sqrt3=-4/3sqrt3`.

Explanation:

`tan270^@` is undefined.

In fact, `tan300^@-tan30^@=tan(360^@-60^@)-1/sqrt3`,

`=-tan60^@-1/sqrt3=-sqrt3-1/sqrt3=-4/sqrt3=-4/3sqrt3`.