$P(\alpha ,\beta )Q(\gamma ,\delta )$ are two point lie on curve ${\tan }^2(x+y)+{\cos }^2(x+y)+y^2+2y=0$ on xy plane. If d=PQ then cos d=

$$\begin{gathered} {\tan }^2(\mathrm{x}+\mathrm{y})+{\cos }^2(\mathrm{x}+\mathrm{y})+{\mathrm{y}}^2+2\mathrm{y}=0 \\ {\sec }^2(\mathrm{x}+\mathrm{y})-1+{\cos }^2(\mathrm{x}+\mathrm{y})+{\mathrm{y}}^2+2\mathrm{y}=0 \\ {\sec }^2(\mathrm{x}+\mathrm{y})+{\cos }^2(\mathrm{x}+\mathrm{y})+1+{\mathrm{y}}^2+2\mathrm{y}=2 \\ 1+{\mathrm{y}}^2+2\mathrm{y}=0 \Rightarrow \mathrm{y}=-1 \\ \mathrm{x}+\mathrm{y}=0 \Rightarrow \mathrm{x}=1 \end{gathered}$$

P and Q are Coincident then d=0

$$\begin{gathered} cos d=1 \\ \mathrm{x}+\mathrm{y}=\mathrm{\pi }, \mathrm{y}=-1 \\ \Rightarrow \mathrm{x}=\mathrm{\pi }+1 \end{gathered}$$

Similarly, if $\mathrm{x}+\mathrm{y}=2\mathrm{\pi }, \mathrm{y}=1$

$\begin{array}{rcl}\mathrm{x}& =& 2\mathrm{\pi }+1\\ \mathrm{P}(\mathrm{\pi }+1, -1), \mathrm{Q}& =& (2\mathrm{\pi }+1, -1)\\ \mathrm{d}& =& \mathrm{PQ}=\mathrm{\pi }\\ \cos \mathrm{d}& =& -1\end{array}$

from (1) and (2) $\cos \mathrm{d}={(-1)}^{\mathrm{n}}, \mathrm{n}\in \mathrm{Z}$