Substituting sine function to cosine function in Bessel function

Question

Suppose that the Bessel function is defined as: $$ J_{n}(\beta)=\frac{1}{2\pi}\int_{0}^{2\pi}e^{j(\beta\,sin\,u-nu)}du $$ Does the following integral have any relationship to the Bessel function? $$ \frac{1}{2\pi}\int_{0}^{2\pi}e^{j(\beta\,cos\,u-nu)}du $$

Answer 1

$$\frac{1}{2\pi}\int_{0}^{2\pi}\exp\Big(j~\big(\beta\,\cos\,u-nu\big)\Big)~du~=~j^n~J_n\big(\beta\big).$$