Given that x and y satisfy the equation: \arctan(x)+\arctan(y)+\arctan(xy)=11/12\pi

Layla-Rose Ellison

Layla-Rose Ellison

Answered question

2022-02-26

Given that x and y satisfy the equation:
arctan(x)+arctan(y)+arctan(xy)=1112π

Answer & Explanation

surgescasjag

surgescasjag

Beginner2022-02-27Added 10 answers

arctanx+arctany+arctan(xy)=arctan(x+y1xy)+arctan(xy)
=arctan(x+y1xy+xy1(x+y1xy)xy=arctan(x+y+xyx2y21xyx2yxy2)=1112π
So
x+y+xyx2y21xyx2yxy2=tan(1112π)
=tanπ12
=tan(π4π3)
x+y+xyx2y2=(1xyx2yxy2)tan(π4π3)
To differentiate both sides you need the chain rule and the product rule, but not the quotient rule. And the tangent of that difference of two fractions is easy to find.

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