In this post, a method to find approximate analytical expression for tangent of every integer in degrees is described. This post does not make any claims about optimality or implementation efficiency for the method described. It is just a fun fact.

All the angles mentioned are in degrees.

Analytical expressions for tan (30), tan (18) and tan (15) are already known. Approximate analytical expression for tan (37) is also known. Using these expressions, we can find tan (1). Once we know tan (1), tangent of any other integer between -90 degrees and 90 degrees can be found out using tan (A +B) formula by writing Tan (N) = Tan (1 + 1+ 1 +.......N times).

Let us look at how to find tan (1).

Since, we know tan (37) and and tan (30), we can find analytical expression for tan (37 - 30) = tan (7).

Since, we know tan (18) and and tan (15), we can find analytical expression for tan (18 - 15) = tan (3).

Since, we know tan (3) and and tan (7), we can find analytical expression for tan (7 - 3) = tan (4).

We also know how to find tan (A\2) in terms of tan (A). Using tan (4), we can find tan (2) and hence, tan (1).