Tan 75/2 solve this

Answer 1

tan[(90-15)/2]

tan [ 45 - 15/2 ]
but tan(a-b) = [tana-tanb]/[1+tana tan b]
so

[1-tan7.5]/[1+tan7.5]

what is tan 7.5 = tan 15/2 = tan 30/4
tan 15/2 = (1-cos 15)/sin 15
but
cos 15 = cos30/2 = sqrt[(1+cos30) /2]
we know cos 30 = sqrt(3)/2
so cos 15 = sqrt[(1+sqrt3/2)/2]
and
sin 15 = sin(30/2) = sqrt[(1-cos30)/2
so sin 15 = sqrt[(1-sqrt3/2)/2
so
tan 7.5 = {1-sqrt[(1+sqrt3/2)/2]}/ {sqrt[(1-sqrt3/2)/2}

now do
(1 -tan 7.5) / (1+tan 7.5)

Answer 2

tan 75 = tan(30+45)
= (tan30+tan45)/(1-tan30*tan45)
= (1/√3 + 1)/(1-1/√3*1)
= (√3+1)/(√3-1)
= (3+2√3+1)/2
= 2+√3

So, now you can figure
sin(75/2) and cos(75/2) and thus tan(75/2)