Two different answers with applying the Cauchy integral formula and parametrizing.

Question

I have the integral $\int_{|z|=1}\frac{cosz}{z}dz$ By applying the Cauchy Integral Formula, I get that this equals $2\pi i*cos(0) = 2\pi i$ Is this correct? If I parametrize the integral with $z= e^{it} $ for $t\in [0,2\pi]$ We have $\int_0^{2\pi}\frac{cos(e^{it})}{e^{it}}dt=0$ I'm not sure which has the mistake.