$\"\"$

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Let $\"\"$ be a real number and $\"\"$ be the point on the unit circle that corresponds to $\"\"$ and $\"\"$ , then $\"\"$ , $\"\"$ , $\"\"$ , $\"\"$ , $\"\"$ , and $\"\"$ .

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$\"\"$

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Consider the equation $\"\"$ .

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Solve for $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$ .

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Thus, the point $\"\"$ .

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That is, for any real number $\"\"$ , there is a point $\"\"$ on the unit circle for which $\"\"$ .

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In other words, the range of the cotangent function is the set of all real numbers.

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$\"\"$

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The range of the cotangent function is the set of all real numbers.