Total no. of orderd pairs (x,y) satisfying `x(sin^2 x+ 1/x^2)=2sinx,sin^2y,` where `x in (-pi,0)uu(0,pi) and y in[0,2pi]` is/are

Answer

8

Explanation

`x(sin^(2) x+1/x^(2))=2 sin x sin^(2) y`
`rArr sin^(2) x+1/x^(2)=2 ((sin x)/x) sin^(2) y`
`rArr sin^(2) x+1/x^(2) le 2 ((sin x)/x)`
But using `A.M. ge G.M.`, we have
`sin^(2) x+1/x^(2) ge 2 (sin x)/x`
`rArr sin^(2) x=1/x^(2)`
or `|sin x|=|1/x|`
Draw the graph of `y=|sin x|` and `y=|1/x|` as shown in the following figure.

From the figure, we get the four values of x.
`sin^(2) y=1` gives `y=pi/2, (3pi)/2`.
so, there will be `4xx2=8` ordered pairs.