Answer
$(1,y)$ and $(x,1)$ for all $x$ and $y$
Work Step by Step
Recall: The tangent to the curve is horizontal when $\frac{dy}{dt}=0$ while it is vertical when $\frac{dx}{dt}=0$.
Find the parameters $\theta$ on which the curve has the horizontal or vertical tangent:
$\frac{dy}{d\theta}=0$ or $\frac{dx}{d\theta}=0$
$\frac{d}{d\theta}(e^{\cos\theta})=0$ or $\frac{d}{d\theta}(e^{\sin\theta})=0$
$-\sin \theta e^{\cos\theta}=0$ or $\cos\theta e^{\sin\theta}=0$
$\sin \theta =0$ or $\cos\theta =0$
Find the points $(x,y)$ on which the curve has the horizontal or vertical tangent:
$x=e^{\sin \theta}=e^0=1$ or $y=e^{\cos\theta}=e^0=1$
Thus, the curve has the horizontal or vertical tangent at the points $(1,y)$ or $(x,1)$.
