There are two cool half angle formulas for tangent:
# tan(theta/2) = {sin theta}/{1 + cos theta}= {1 - cos theta}/{sin theta} #
We know
#cos(-30^circ) = \sqrt{3}/2 #
#sin(-30^circ) = -1/2 #
Looks like the one with sine in the denominator will be best:
#tan(-15^circ) = {1 - cos(-30^circ) }/{ sin(- 30^circ)} = {1 - \sqrt{3}/2}/{-1/2} = sqrt{3}-2 #
Let's check that by calculating sine and cosine with the difference angle formulas.
# cos(-15)=cos(30-45)=cos 30 cos 45 + sin 30 sin 45 = \sqrt{2}/4(\sqrt{3} + 1) #
# sin(-15)=sin(30-45)=sin 30 cos 45 - cos 30 sin 45 = \sqrt{2}/4(1
- \sqrt{3}) #
#tan(-15) = sin(-15)/cos(-15) = (1 - \sqrt{3})/(1+\sqrt{3}) times (1-sqrt{3})/(1-sqrt{3}) ={4-2sqrt{3}}/{-2} =sqrt{3}-2 quad sqrt #
