tan75:

method (1):

tan75 = sin75/cos75

= sin(45+30)/cos(45+30)

= (sin45cos30+sin30cos45)/(cos45cos30 - sin45sin30)

= [sqrt2/2*sqrt3/2 + (1/2)*sqrt2/2]/(sqrt2/2*sqrt3/2 - sqrt2/2*(1/2)

= (sqrt(6)/6 + sqrt2/4)/(sqrt6/4)- sqrt2/4)

= (2sqrt6 + 3sqrt2)/12 / (sqrt6/4 - sqrt2/4)

= (2sqrt6 + 3sqrt2)/12 / (sqrt6-sqrt2)/4

= (2sqrt6 + 3sqrt2)/3(sqrt6-sqrt2)

Multiply by sqrt6 + sqrt2

==> (sqrt6+sqrt2)(2sqrt6+3sqrt2) / 3(6-2)

==> (2*6 + 3*2sqrt3 + 2*2sqrt3 + 3*2)/12

==> (18 + 10sqrt3)/12

==> (9+ 5sqrt3)6

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