If `f(x , y)` satisfies the equation `1+4x-x^2=sqrt(9sec^2y+4cos e c^2y)` then find the value of `xa n dtan^2ydot`

Explanation

We have
`1+4x-x^2=sqrt(9sec^4y+4cosec^2y)`
Now, `1+4x-x^2=5-(x-2)^2le5`
`9sec^2y+4cosec^2y=13+9tan^2y+4cot^2y`
`ge(3tany-2coty)^2+25ge25`
`sqrt(9sec^2y+4cosec^2y)ge5`
Hence, `1+4x-x^2=sqrt(9sec^4y+4cosec^2y)` is possible only when
`1+4x-x^2=sqrt(9sec^4y+4cosec^2y)=5`
`rArr x=2" and " 9tan^2yy+4cot^2y=12`
`rArr x=2" and "tan^2y=2/3`