That looks good to me. .
The tangent will be horizontal when the numerator is 0:
[tex]cos(\theta)- 2cos^2(\theta)= cos(\theta)(1- 2cos(
\theta))= 0. That will be true wnen [tex]cos(\theta)= 0[tex] or when .
The tangent will be vertical when the denominator is 0:
when or .
Since both numerator and denominator are 0 when , that is not necessarily a vertical or horizontal tangent. You will need to look more closely at that.