So I have this identity or result from studying polygons and complex numbers which states:
$$\prod_{j=0}^{n-1}\sqrt{\bigg(\cos\frac{\theta+2j\pi}{n}-\cos\frac{\theta}{n}\bigg)^2 +\bigg(\sin\frac{\theta+2j\pi}{n}-\sin\frac{\theta}{n}\bigg)^2}=n $$
I know how to prove this directly using trigonometric identities, and roots of unity of complex numbers, which is quite a lengthy process in my opinion. But my question is, how do I prove this using induction?
For instance going to the induction step straight way give us:
Step I: Let $n = k,$ if $P(k)$ holds true then for $P(k+1)$ we have: $$P(k)=\prod_{j=0}^{k-1}\sqrt{\bigg(\cos\frac{\theta+2j\pi}{k}-\cos\frac{\theta}{k}\bigg)^2 +\bigg(\sin\frac{\theta+2j\pi}{k}-\sin\frac{\theta}{k}\bigg)^2}=k $$ $$P(k+1)=\prod_{j=0}^{k}\sqrt{\bigg(\cos\frac{\theta+2j\pi}{k+1}-\cos\frac{\theta}{k+1}\bigg)^2 +\bigg(\sin\frac{\theta+2j\pi}{k+1}-\sin\frac{\theta}{k+1}\bigg)^2} $$
I am not sure if I am doing this correctly. Furthermore I do not know what to do next after the induction step like how it is carried out in normal induction. I would appreciate some guidance or a possible root to carry out the induction on this result.
