Use the fact that tangent is an odd function. This means
#tan(-pi/12) = -tan(pi/12)#
Use the Half Angle Identity #-tan(theta/2) = -sqrt((1-cos theta)/(1+cos theta))#:
Let #theta = pi/6 => tan((pi/6)/2) = tan(pi/6 * 1/2) = tan(pi/12)#
# -tan((pi/6)/2) = -sqrt((1-cos( pi/6))/(1+cos (pi/6))#
#cos (pi/6) = cos (30^@) = sqrt(3)/2#
# -tan((pi/6)/2) = -sqrt((1- sqrt(3)/2)/(1 + sqrt(3)/2))#
Find common denominators:
#tan(-pi/12) = -sqrt(((2-sqrt(3))/2)/((2+sqrt(3))/2)) = - sqrt(((2-sqrt(3))/cancel(2)) * cancel(2)/(2+sqrt(3)))#
#tan(-pi/12) = -sqrt((2-sqrt(3))/(2+sqrt(3))#
Multiply by the conjugate:
#tan(-pi/12) = -sqrt((2-sqrt(3))/(2+sqrt(3)) * (2-sqrt(3))/(2-sqrt(3))#
#tan(-pi/12) = - sqrt((2-sqrt(3))^2/(4 - 3)) = - (2-sqrt(3))#
#tan(-pi/12) =sqrt(3) - 2#
