Here is a cool trigonometric identity I recently encountered:
For example, for :
After thinking about it for some time I realised that the terms on the left side can each be seen as half the lengths of chords of a unit circle with 4n evenly spaced points that can then be rearranged to be distances from a point on the unit circle to half of the points of a regular (2n)-gon, as shown in the figure below.
With the insight of this figure we then write
(The cosine formula can be derived in a similar manner.)
In general, the product of the distances of any point in the complex plane to the n roots of unity is
The above case was where . Two more cases are illustrated below, this time for n = 10. In the left example the product of distances is
while for the right example it is
Note that earlier in the year I posted on the distances to a line from vertices of a regular polygon.