Please confirm my answer for this trigonometric identity?

Question

So I finally figured this out I think. Here is the problem:

If $f(\theta)= cot(\theta)= -2$, find $f( \theta + \pi)$.

So I drew a circle since $-2$ is at about $-115$ degrees I set my point there and then rotated it $180$ degrees in the positive direction which lands me in the second quadrant with a negative cot. Thus, the answer is $-2$ correct?

Answer 1

An alternative way to solving the problem is to observe \begin{align} f(\theta) = \frac{\cos\theta}{\sin\theta} \end{align} which means \begin{align} f(\theta + \pi) = \frac{\cos(\theta+\pi)}{\sin(\theta+\pi)}. \end{align} Now use the angle sum formula for sine and cosine.