Calculating the equation of the plane tangent to a given surface in xyz space.

Question

The solution is written below and I understood all the solution until saying that the value of z is 1, but I do not understand the rest, why z + x = 1, could anyone clarify this for me please ?

Answer 1

The gradient of the surface is given by $\nabla = \langle -e^{-x}siny, e^{-x}cosy \rangle $. Evaluating this gradient at $x = 0, y = \frac{\pi}{2}$, you find $\langle (-e^{0})sin(\frac{\pi}{2}), e^{0}cos(\frac{\pi}{2}) \rangle = \langle -1, 0 \rangle$. Therefore the slope of $x = -1, y = 0$. The value of $z$ at this surface is $1$ so you get $z = -x + 1$ or $z + x = 1$ as your tangent plane.