I expected the following formula to hold:


$\int^{2n\pi}\_0\cos(\sin t+t/n)dt=0$,
for ${}^\forall n\in\mathbb{N},\ n\geq2$


But I can't prove it.
Could you please tell me.



An integral of composite function of triangle functions
You can rewrite the integral as
$$
\int\_0^{2\pi} \left(\sum\_{j=0}^{n-1}\cos\Big(\sin t+\tfrac tn+2\pi \tfrac jn\Big)\right)\,dt.
$$
But $\sum\_{j=0}^{n-1}\cos\big(a+2\pi \tfrac jn\big)=0$ for all $a$.


In particular, the equality holds if $\sin t$ is replaced by *any* $2\pi$-periodic function.


