You have show that the condition for stationary points is that tanx = 1
In the next part we actually have to find those points,
that is, solve
tanx = 1
x = 45° or x = 225°
or
x = π/4 or x = 5π/4 radians
when x = π/4
y = e^-π/4 sin π/4 = .3224
when x = 5Ï€/4
y = e^-5Ï€/4 sin 5Ï€/4 = -.011139
Looking at your domain, I think you made a typo, and I will assume you meant
-π < x < π
so the second point above lies outside the domain,
but there is another solution of
x = -135° or -3π/4
if x = -3Ï€/4
y = e^-3Ï€/4 sin 3Ï€/4 = -.067
Within your domain, the stationary points are
(Ï€/4, .3224) and (-3Ï€/4, -.067)