Prove the trigonometric identities

by the following group-theoretical means.
1. Show that the complex numbers of unit modulus \(c=x+i y\) with \(|x|^{2}+|y|^{2}=1\) form a group under multiplication, and that \(e^{i \phi}\) with \(-\pi \leq \phi \leq \pi\) is a faithful representation: \(\cos \phi+i \sin \phi=e^{i \phi}\).
2. Use the representation \(e^{i \phi}\) and group multiplication to prove the identities.
This is a simple example of a more general idea: the special functions of mathematical physics often are representations of some group, and standard identities can be obtained by appropriate operations on the representations of that group.

Transcribed Image Text:

cos(+8)= cos o cos sin(+0) = cos o sin - sin o sin + sin cos