Question

How do you find the exact value: `(tan 325° - tan 25°) / (1 + tan 325°(tan 25°))`?

Answer 1

`-sqrt3`

Explanation:

Note that this is the tangent angle subtraction formula in reverse:

`tan(a-b)=(tana-tanb)/(1+tanatanb)`

Thus:

`tan(325˚-25˚)=(tan325˚-tan25˚)/(1+tan325˚(tan25˚))=tan(300˚)`

Note that `300˚` is in the fourth quadrant, where tangent is negative. `300˚` also has a reference angle of `60˚`, so we see that `tan(300˚)=-tan(60˚)=-sqrt3`.