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## Homework Statement

Prove that ##\tan (1^\circ)## is irrational.

## Homework Equations

## The Attempt at a Solution

Suppose for contradiction that ##\tan (1^\circ)## is rational. We claim that this implies that ##\tan (n^\circ)## is rational. Here is the proof by induction: We know by supposition that the base case holds. So, suppose that ##\tan (n^\circ)## is rational. Then ##\displaystyle \tan (n^\circ + 1^\circ) = \frac{\tan(n^\circ) + \tan(1^\circ)}{1-\tan(n^\circ)\tan(1^\circ)}##, and this is the ratio of two rational numbers, and so is rational. So by mathematical induction ##\tan (n^\circ)## is rational.

However, this implies that ##\tan(30^\circ) = \sqrt{3}## is rational, which is a contradiction.