If we can construct the lengths `cos alpha` or `sin alpha`, can we then construct `tan alpha`? How about the converse, if `tan alpha` is constructible, is `cos alpha` or `sin alpha` constructible?
First assume we are always given the length 1. If we can construct the length of `cos alpha`, then we can construct the length `sin alpha` and vice versa, since they form the sides of a right triangle with hypoteneous `1`.
What about tangent? Indeed, if we can construct `cos alpha`, then we can construct `tan alpha` as it is the ratio of sine to cosine.
Conversely, if we can construct `tan alpha`, then by drawing a perpendicular segment of length `1` to the edge of the length `tan alpha`, we create a right triangle of an angle `alpha`. To which we can then draw a hypoteneous of length `1` to get both `cos alpha` and `sin alpha`. In conclusion
Given the length 1, the length `cos alpha` is constructible if and only if `sin alpha` is constructible if and only if `tan alpha` is constructible.
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