Question

How do you prove that `tan15=2-sqrt3`?

Answer 1

Below

Explanation:

RTP: `tan15=2-sqrt3`

`tan15 = tan(45-30)`

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

`tan(45-30)=(tan45-tan30)/(1+tan45tan30)`

= `(1-1/sqrt3)/(1+1/sqrt3)`

= `((sqrt3-1)/sqrt3)/((sqrt3+1)/sqrt3)`

= `((sqrt3-1)/sqrt3)times(sqrt3/(sqrt3+1))`

= `(sqrt3-1)/(sqrt3+1)`

Rationalise the denominator
= `(sqrt3-1)/(sqrt3+1)times(sqrt3-1)/(sqrt3-1)`

= `(3-2sqrt3+1)/(3-1)`

= `(4-2sqrt3)/2`

Take out the common factor
= `(2(2-sqrt3))/2`

Simplify
= `2-sqrt3`