Prove that each singular point of tan z is a simple pole

Question : Prove that each singular point of tan z is a simple pole. Solution : Singularities of tan z Since we can write \[tan\ z\ =\frac{\sin{z}}{\cos{z}}=\frac{p\left(z\right)}{q\left(z\right)}\] Singularities of tan z are the zeros of cos z \[\cos{z}=0\] \[\cos{z}=\cos{\frac{n\pi}{2}},\ n=\pm1,\pm3,\pm5\ldots..…

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