Answer
${{\bf{T}}_\theta }{|_{\left( {\frac{{\rm{\pi }}}{2},\frac{{\rm{\pi }}}{4}} \right)}} = \left( { - \frac{1}{2}\sqrt 2 ,0,0} \right)$
${{\bf{T}}_\phi }{|_{\left( {\frac{{\rm{\pi }}}{2},\frac{{\rm{\pi }}}{4}} \right)}} = \left( {0,\frac{1}{2}\sqrt 2 , - \frac{1}{2}\sqrt 2 } \right)$
${\bf{N}}\left( {\frac{{\rm{\pi }}}{2},\frac{{\rm{\pi }}}{4}} \right) = - \frac{1}{2}{\bf{j}} - \frac{1}{2}{\bf{k}}$
The equation of the tangent plane:
$y + z = \sqrt 2 $
Work Step by Step
We have $G\left( {\theta ,\phi } \right) = \left( {\cos \theta \sin \phi ,\sin \theta \sin \phi ,\cos \phi } \right)$. So,
${{\bf{T}}_\theta } = \frac{{\partial G}}{{\partial \theta }} = \left( { - \sin \theta \sin \phi ,\cos \theta \sin \phi ,0} \right)$, ${\ \ \ }$ ${{\bf{T}}_\theta }{|_{\left( {\frac{{\rm{\pi }}}{2},\frac{{\rm{\pi }}}{4}} \right)}} = \left( { - \frac{1}{2}\sqrt 2 ,0,0} \right)$
${{\bf{T}}_\phi } = \frac{{\partial G}}{{\partial \phi }} = \left( {\cos \theta \cos \phi ,\sin \theta \cos \phi , - \sin \phi } \right)$, ${\ \ \ }$ ${{\bf{T}}_\phi }{|_{\left( {\frac{{\rm{\pi }}}{2},\frac{{\rm{\pi }}}{4}} \right)}} = \left( {0,\frac{1}{2}\sqrt 2 , - \frac{1}{2}\sqrt 2 } \right)$
${\bf{N}}\left( {\frac{{\rm{\pi }}}{2},\frac{{\rm{\pi }}}{4}} \right) = {{\bf{T}}_\theta } \times {{\bf{T}}_\phi } = \left| {\begin{array}{*{20}{c}}
{\bf{i}}&{\bf{j}}&{\bf{k}}\\
{ - \frac{1}{2}\sqrt 2 }&0&0\\
0&{\frac{1}{2}\sqrt 2 }&{ - \frac{1}{2}\sqrt 2 }
\end{array}} \right|$
${\bf{N}}\left( {\frac{{\rm{\pi }}}{2},\frac{{\rm{\pi }}}{4}} \right) = - \frac{1}{2}{\bf{j}} - \frac{1}{2}{\bf{k}}$
The equation of the tangent plane to the surface at the point $G\left( {\frac{{\rm{\pi }}}{2},\frac{{\rm{\pi }}}{4}} \right) = \left( {0,\frac{1}{2}\sqrt 2 ,\frac{1}{2}\sqrt 2 } \right)$ is
$\left( {x,y - \frac{1}{2}\sqrt 2 ,z - \frac{1}{2}\sqrt 2 } \right)\cdot\left( {0, - \frac{1}{2}, - \frac{1}{2}} \right) = 0$
$ - \frac{1}{2}\left( {y - \frac{1}{2}\sqrt 2 } \right) - \frac{1}{2}\left( {z - \frac{1}{2}\sqrt 2 } \right) = 0$
$y - \frac{1}{2}\sqrt 2 + z - \frac{1}{2}\sqrt 2 = 0$
The equation can be written as
$y + z = \sqrt 2 $
