Real part of e^ei^theta is

(1) e^{cos θ}( cos (sin θ)

(2) e^{cos θ}( cos (cos θ)

(3) e^{sin θ}( sin (cos θ)

(4) e^{sin θ}( sin (sin θ)

Solution:

e^{cos θ} (e^{i sin θ}) = e^{cos θ}( cos sin θ) + ie^{cos θ} ( sin sin θ)

The real part is e^{cos θ}( cos sin θ)

Hence option (1) is the answer.