The number of points `P(x , y)` lying inside or on the circle `x^2+y^2=9` and satisfying the equation `tan^4x+cot^4x+2=4sin^2y` is______

Answer

8

Explanation

We have `tan^(4)x +cot^(4)x+2=4 sin^(2)y`
or `( tan^(4)x-cot ^(2)x)^(2)+4=4 sin^(2)y`
Now, `LHS ge 4` and RHS `le4` . Therefore,
`tan^(2)x=1` and `sin^(2)y=1` or `tan x = 1` and `sin^(2)y=1` or `tan x = +- 1` and `sin y = +1`
But ` -3 lex le 3` and `-3 le y le3`.
Therefore, the acceptable values of x are` +- pi //4` and `+- 3pi //4`. The acceptable values of y are `+- pi //2`.
Hence, the number of points `P( x,y)` are 8.