Question

Cot(x+iy) separate real and inaginery parts

Answer 1

SOLUTION

TO DETERMINE

To separate real and Imaginary parts

cot(x + iy)

EVALUATION

We know that

$\sf{ \sin (x + iy) = \sin x \cosh y + i \cos x \sinh y}$

$\sf{ \cos (x + iy) = \cos x \cosh y - i \sin x \sinh y}$

$\sf{ \therefore \: \: \cot (x + iy) }$

$\displaystyle \: \sf{ = \frac{ \cos (x + iy)}{ \sin (x + iy)} }$

$\displaystyle \: \sf{ = \frac{ \cos x \cosh y - i \sin x \sinh y}{ \sin x \cosh y + i \cos x \sinh y} }$

$\displaystyle \: \sf{ = \frac{ \sin 2 x - i \sinh 2y}{ \cosh 2y - \cos 2x } }$

$\displaystyle \: \sf{ = \frac{ \sin 2 x }{ \cosh 2y - \cos 2x } - i \displaystyle \: \sf{ \frac{ \sinh 2y}{ \cosh 2y - \cos 2x } }}$

Real part

$\displaystyle \: \sf{ = \frac{ \sin 2 x }{ \cosh 2y - \cos 2x } }$

Imaginary part

$\displaystyle \: \sf{ = - \displaystyle \: \sf{ \frac{ \sinh 2y}{ \cosh 2y - \cos 2x } }}$

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