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Question

Seperate tan1(x+iy) in to real and imaginary parts.

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Solution

Let α+iβ=tan1(x+iy)
Then αiβ=tan1(xiy)
Adding, we get
2α=tan1(x+iy)+tan1(xiy)
tan1(x+iy+xiy1(x+iy)(xiy))
Real part α=12tan1(x+iy+xiy1(x+iy)(xiy))
Subtracting, 2iβ=tan1(x+iy)tan1(xiy)
2iβ=tan1((x+iy)(xiy)1+(x+iy)(xiy))
2iβ=tan1(2iy1+x2+y2)
2iβ=itanh1(2y1+x2+y2)
Imaginary part β=12tanh1(2y1+x2+y2)

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