The number of points P(x,y) lying inside or on the circle ${\mathrm{x}}^2+{\mathrm{y}}^2=9$ and satisfying the equation ${\tan }^4\mathrm{x}+{\cot }^4\mathrm{x}+2=4{\sin }^2\mathrm{y}$  is

We have ${\tan }^4\mathrm{x}+{\cot }^4\mathrm{x}+2=4{\sin }^2\mathrm{y}$

or  ${({\tan }^2\mathrm{x}-{\cot }^2\mathrm{x})}^2+4=4{\sin }^2\mathrm{y}$

Now, LHS $\ge$ 4 and RHS $\le$ 4. Therefore,

${\tan }^2\mathrm{x}=1 \text{and }{\sin }^2\mathrm{y}=1 \text{or }\tan \mathrm{x}=\pm 1 \text{and }\sin \mathrm{y}=\pm 1$

but, $-3\le \mathrm{x}\le 3 \text{and }-3\le \mathrm{y}\le 3$

Therefore, the acceptable values of x are $\pm \frac{\mathrm{\pi }}{4}$and $\frac{\pm 3\mathrm{\pi }}{4}$. The
acceptable values of y are $\pm \frac{\mathrm{\pi }}{2}$

Hence, the number of points p(x,y) are 8