tan(195^o) = tan(390^o/2)
The identity you need is:
tan(x/2) = pmsqrt((1-cosx)/(1+cosx))
+ if quadrant I or III
- if quadrant II or IV
It's derived from:
sin(x/2) = pmsqrt((1-cosx)/2)
+ if quadrant I or II
- if quadrant III or IV
cos(x/2) = pmsqrt((1+cosx)/2)
+ if quadrant I or IV
- if quadrant II or III
Divide the quadrant conditions to get the sign you need. tan390^o = tan30^o, so you're in quadrant I. Thus, no matter what, the sign of the answer is positive.
tan(390^o/2) = sqrt((1-cos390^o)/(1+cos390^o))
= sqrt((1-cos30^o)/(1+cos30^o))
= sqrt((1-sqrt3/2)/(1+sqrt3/2))
Get common denominators:
= sqrt(((2-sqrt3)/2)/((2+sqrt3)/2))
Cancel:
= sqrt((2-sqrt3)/(2+sqrt3))
Multiply by the conjugate of the denominator:
= sqrt(2-sqrt3)/sqrt(2+sqrt3)*sqrt(2-sqrt3)/sqrt(2-sqrt3)
Simplify:
= (2-sqrt3)/sqrt(2^2-(sqrt3)^2)
= (2-sqrt3)/sqrt(1)
= 2-sqrt3