Answer
$$e - 1$$
Work Step by Step
$$\eqalign{
& \int_0^{\pi /2} {\int_0^{\cos y} {{e^{\sin y}}} dx} dy \cr
& {\text{Integrate with respect to }}x \cr
& = \int_0^{\pi /2} {\left[ {x{e^{\sin y}}} \right]_0^{\cos y}dy} \cr
& {\text{Evaluating the limits}} \cr
& = \int_0^{\pi /2} {\left( {\cos y{e^{\sin y}} - 0{e^{\sin y}}} \right)dy} \cr
& = \int_0^{\pi /2} {{e^{\sin y}}\cos ydy} \cr
& {\text{Integrate}} \cr
& = \left[ {{e^{\sin y}}} \right]_0^{\pi /2} \cr
& {\text{Evaluating}} \cr
& = {e^{\sin \left( {\pi /2} \right)}} - {e^{\sin 0}} \cr
& = {e^1} - {e^0} \cr
& = e - 1 \cr} $$
