Can someone please explain to me why they can transform in the blue circle? from 1 to 2

 

I recognize it as a phase shift but my grasp of trig is too long ago to explain it The original equation appears to be describing a sawtooth-like pulse, though its not clear if that's the stimulus or the result...


However I find, ChatGPT is a good source, with known caveats, of explaining these things, so here's its explanation after asking it to compare images of the two equations..


\[ v(t) = 1 + \sum_{n=1}^{\infty} \frac{2(-1)^n}{1 + n^2} (\cos nt - n \sin nt) \] Using the trigonometric identity: \[ \cos nt - n \sin nt = \sqrt{1 + n^2} \cos \left( nt + \tan^{-1} n \right), \] we rewrite the summation term as: \[ \frac{2(-1)^n}{1 + n^2} (\cos nt - n \sin nt) = \frac{2(-1)^n}{1 + n^2} \cdot \sqrt{1 + n^2} \cos \left( nt + \tan^{-1} n \right). \] Simplifying the coefficient: \[ \frac{2(-1)^n}{1 + n^2} \cdot \sqrt{1 + n^2} = \frac{2(-1)^n}{\sqrt{1 + n^2}}. \] Thus, the transformed equation becomes: \[ v(t) = 1 + \sum_{n=1}^{\infty} \frac{2(-1)^n}{\sqrt{1 + n^2}} \cos \left( nt + \tan^{-1} n \right). \] This confirms that the new equation is simply a phase-shifted version of the original sum.


I'm sure someone will argue that I shouldn't propose using ChatGPT or that this is 'too much help, go do your research' but I disagree... this is a known substitution that can be found in many textbooks if you know where to look...