Tan(-15) Find exact value using difference identity

Thank you.

tan(-15)
= tan(30 - 45)
Using the identity:
tan(u - v) = (tan(u) - tan(v)) / (1 + tan(u)tan(v))
tan(30 - 45) = (tan(30) - tan(45)) / (1 + tan(30)tan(45))
since tan(30) = 1/sqrt(3) and tan(45) = 1
tan(30 - 45) = ((1/sqrt(3)) - 1) / (1 + (1/sqrt(3))(1))
= (1/sqrt(3) - sqrt(3)/sqrt(3)) / (sqrt(3)/sqrt(3) + 1/sqrt(3))
= ((1 - sqrt(3))/sqrt(3)) / ((1 + sqrt(3))/sqrt(3))
Multiply numerator and denominator by sqrt(3):
= (1 - sqrt(3)) / (1 + sqrt(3))
Multiply numerator and denominator by (1 - sqrt(3))
since (a + b)(a - b) = a^2 - b^2:
= (1 - sqrt(3))^2 / (1 - 3)
= (1 - 2sqrt(3) + 3) / (-2)
= (4 - 2sqrt(3)) / (-2)
= sqrt(3) - 2