91Ó°ÊÓ

Given $f(x,y)=\tan (x+y)$at point $P=\left(x_0,y_0\right)=(0,\mathrm{Ï€})$

The equation of line of tangent is

$$\begin{gathered} f_x\left(x_0,y_0\right)\left(x-x_0\right)+f_y\left(x_0,y_0\right)\left(y-y_0\right)=f\left(x_0,y_0\right) \\ {f}_x\left(0,\mathrm{Ï€}\right)\left(x-0\right)+f_y\left(0,\mathrm{Ï€}\right)\left(y-\mathrm{Ï€}\right)=z-f\left(0,\mathrm{Ï€}\right)\left(1\right) \end{gathered}$$

$f_x\left(x_0,y_0\right)={\left.\frac{d}{dx}f(x,y)\right|}_{\left(x_0,y_0\right)}={\left.\frac{d}{dx}\tan (x+y)\right|}_{\left(x_0,y_0\right)}$

$f_x\left(x_0,y_0\right)={\left.\frac{d}{dx}\left(\tan \right(x+y\left)\right)\frac{d}{dx}(x+y)\right|}_{\left(x_0,y_0\right)}$

$f_x\left(x_0,y_0\right)={\left.{\sec }^2(x+y)·1\right|}_{\left(x_0,y_0\right)}$

$f_x\left(x_0,y_0\right)={\left.{\sec }^2(x+y)\right|}_{\left(x_0,y_0\right)}$

$f_x(0,\mathrm{Ï€})={\left.{\sec }^2(x+y)\right|}_{(0,\mathrm{Ï€})}$

$f_x(0,\mathrm{Ï€})={\sec }^2(0+\mathrm{Ï€})$

$f_x(0,\mathrm{Ï€})={\sec }^2\left(\mathrm{Ï€}\right)$

$f_x(0,\mathrm{Ï€})=\frac{1}{{\cos }^2\left(\mathrm{Ï€}\right)}=\frac{1}{\left(\cos \right(\mathrm{Ï€}){)}^2}$

(∵cos(Ï€)=-1)$f_x(0,\mathrm{Ï€})=\frac{1}{(-1{)}^2}$ $(âˆ\mu \cos (\mathrm{Ï€})=-1)$

$f_x(0,\mathrm{Ï€})=1$ $\left(2\right)$

$f_y\left(x_0,y_0\right)={\left.\frac{d}{dy}f(x,y)\right|}_{\left(x_0,y_0\right)}={\left.\frac{d}{dy}\tan (x+y)\right|}_{\left(x_0,y_0\right)}$

$f_y\left(x_0,y_0\right)={\left.\frac{d}{dy}\left(\tan \right(x+y\left)\right)\frac{d}{dy}(x+y)\right|}_{\left(x_0,y_0\right)}$

$f_y\left(x_0,y_0\right)={\left.{\sec }^2(x+y)·1\right|}_{\left(x_0,y_0\right)}$

$f_y\left(x_0,y_0\right)={\left.{\sec }^2(x+y)\right|}_{\left(x_0,y_0\right)}$

$f_y(0,\mathrm{Ï€})={\left.{\sec }^2(x+y)\right|}_{(0,\mathrm{Ï€})}$

$f_y(0,\mathrm{Ï€})={\sec }^2(0+\mathrm{Ï€})$

$f_y(0,\mathrm{Ï€})={\sec }^2\left(\mathrm{Ï€}\right)$

$f_y(0,\mathrm{Ï€})=\frac{1}{{\cos }^2\left(\mathrm{Ï€}\right)}=\frac{1}{\left(\cos \right(\mathrm{Ï€}){)}^2}$

$f_y(0,\mathrm{Ï€})=\frac{1}{(-1{)}^2}$ $(âˆ\mu \cos \left(\mathrm{Ï€}\right)=-1)$

$f_y(0,\mathrm{Ï€})=1$ $\left(3\right)$

$f\left(x_0,y_0\right)=f(0,\mathrm{Ï€})=\tan (0+\mathrm{Ï€})$

$f(0,\mathrm{Ï€})=\tan \left(\mathrm{Ï€}\right)$

$f(0,\mathrm{Ï€})=0$ $(âˆ\mu \tan (\mathrm{Ï€})=0)\left(4\right)$

$$\begin{gathered} 1(x-0)+1(y-\mathrm{Ï€})=\mathrm{z}-0 \\ \mathrm{x}-0+\mathrm{y}-\mathrm{Ï€}=\mathrm{z} \\ \mathrm{x}+\mathrm{y}-\mathrm{z}=\mathrm{Ï€} \end{gathered}$$