TITLE: How many points to prove a trigonometric identity?
QUESTION [9 upvotes]: I am taking a trigonometric identity from another post, arbitrarily.
$$\frac{2\sec\theta +3\tan\theta+5\sin\theta-7\cos\theta+5}{2\tan\theta +3\sec\theta+5\cos\theta+7\sin\theta+8}=\frac{1-\cos\theta}{\sin\theta}.$$
Besides the usual approach by reworking/simplifying the expressions using elementary identities, one could use a "lazy" approach, by evaluating both members for several $\theta$ and checking equality.
This works for polynomials, if you probe them at $d+1$ points, where $d$ is the degree.
Can we derive general rules about the number of equalities required to guarantee that trigonometric expressions of a certain complexity are indeed identical ?

REPLY [5 votes]: If the numerator and denominator of an trigonometric expression (supposed to equal $0$) are trigonometric polynomials of degree at most $k$ then the fraction is a rational function of $u = \exp(i \theta)$ of degree at most $2k$, and $4k+1$ distinct angles would be needed.
This matches the number of coefficients (minus 1 for scaling) needed to write a ratio of two trigonometric polynomials.
I don't see how one could make exact comparison of the values at so many distinct points without having a separate proof of the identity.  For numerical evaluation the question of accuracy needed is similar to that for rational functions of the same degree and size of coefficients, but maybe there is some simplification from knowing $|u|=1$.