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Suppose that X has a standard uniform distribution. Show that Y=tan(2πX) has a standart Cauchy distribution.

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Let's consider the cumulative distribution function of $Y$: $$ P(Y\leq x) = P(\tan(2\pi X)\leq x). $$ You can probably continue from here to get the CDF of the Cauchy distribution by considering separately the cases where $x$ is positive or negative.

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