How to resolve trigonometric relationships

Question

Is there any generic method of developing the trigonometric relationships of multiple arcs without successively applying the double / triple arc / sum of "formulas"? Like develop $\tan (7x)$ in a fast way? If so, could you advise me some material about it?

Answer 1

Yes, use the tan-half-angle substitution of $t = \tan \left( \tfrac{\theta}{2} \right) $

This allows for the following substitutions

$$ \sin \theta = \frac{2 t}{1+t^2} $$

$$ \cos \theta = \frac{1-t^2}{1+t^2} $$

So any expression it terms of $\sin(\theta)$ and $\cos(\theta)$ is converted into a polynomial in terms of $t$. After $t$ is found, then use $$ \theta = 2 \tan^{-1} (t) $$ to get back to the angle.

See this answer for an example of how to solve a trig expression like $$a \sin(\theta) + b \cos(\theta) = c $$