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.