I'm trying to derive the integral form of the Bessel function by finding the $k$th coefficient of the Laurent series expansion of the function $f(z) =\exp [\lambda(z-\frac{1}{z})]$. I managed to get it down to the form
$ J_k(\lambda) =\frac{1}{2\pi}\int_{0}^{2 \pi} e^{i(\lambda \sin\theta - k \theta)} d\theta =\frac{1}{2\pi}\int_{0}^{2 \pi} [\cos(\lambda \sin\theta - k \theta) +i\sin(\lambda \sin\theta - k \theta)]d\theta $
But, I need to show that this is equivalent to $ \frac{1}{2\pi}\int_{0}^{2 \pi} \cos(\lambda \sin\theta - k \theta)d\theta $. In other words, I need to show that $ \int_{0}^{2 \pi}\sin(\lambda \sin\theta - k \theta)d\theta =0 $
But I can't figure out how to do this. I tried expanding using trig identities and then writing sin and cos as Taylor Series and integrating term by term, but no luck. What am I missing?