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For all, x ,y in Rdot find the range of...

For all, `x ,y in Rdot` find the range of `((x+y)(1-x y))/((1+x^2)(1+y^2))` .

Text Solution

Verified by Experts

Let `x=tan alpha, y=tan beta`
`therefore ((x+y)(1-xy))/((1+x^(2))(1+y^(2)))`
`=((tan alpha+tan beta)(1-tan alpha tan beta))/(sec^(2)alpha sec^(2)beta)`
`=sin(alpha+beta)cos(alpha+beta)`
`=(1)/(2)sin(2alpha+2beta)`
`therefore` Thus, required range is `[(-1)/(2),(1)/(2)]`
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