This question is motivated from the following excerpt from Rational Points on Elliptic Curves by Silverman and Tate:
$$\cos \theta = \frac{1 – t^2}{1 + t^2}, \sin \theta = \frac{2t}{1+t^2}$$
If you have some complicated identity in sine and cosine that you want
to test, all you have to do is substitute these formulas, collect powers of $t$,
and see if you get zero. (If they had told you this in high school, the whole
business of trigonometric identities would have become a trivial exercise in
algebra!)
I realized that they are absolutely right. The above substitution and de'Moivre's formula allows us to convert any polynomial equation in $\left\{\sin n \theta, \cos n\theta \right\}_{n\in \mathbb{N}}$ into a polynomial equation in $t$. And a polynomial equation in $t$ can be simple to verify (may be laborious at times though).
Are there trigonometric formulas in one variable that cannot be derived through this method? If you know any, please point it out.
Additionally, are there software packages that use this method to verify trigonometric formulas? I would like to know any algorithms used to verify trig. identities.
Thank you 🙂
Best Answer
Right now a simple answer is "any trigonometric formula involving more than one variable." The most straightforward fix, to my mind, doesn't involving generalizing this idea but using Euler's formula instead (generalizing this idea requires using the angle addition formula whereas in the Euler's formula approach the angle addition formula is a corollary). That is, instead make the substitutions
$$\cos \theta = \frac{e^{i \theta} + e^{-i \theta}}{2}, \sin \theta = \frac{e^{i \theta} - e^{-i \theta}}{2i}$$
and similarly for any other variables that appear.