Let us take the following equality:
#e^{i theta} = cos theta + i sin theta#
Thus:
#e^{{13 pi}/{12} i} = cos ({13 pi}/{12}) + i sin ({13 pi}/{12})=#
# = -{1 + sqrt{3}}/{2 sqrt{2}} - i {sqrt{3} - 1}/{2 sqrt{2}} = {-1 - sqrt{3} - i sqrt{3} + i}/{2 sqrt{2}} = #
# = {- (1 + sqrt{3}) + i (1 - sqrt{3})}/{2 sqrt{2}}#
In the same way:
#e^{{13 pi}/{2} i} = cos ({13 pi}/{2}) + i sin ({13 pi}/{2})=#
# = 0 + i cdot 1 = i#
We sum it:
#e^{{13 pi}/{12} i} - e^{{13 pi}/{2} i} = {- (1 + sqrt{3}) + i (1 - sqrt{3})}/{2 sqrt{2}} - i = #
#= {- (1 + sqrt{3}) + i (1 - sqrt{3})- i cdot 2 sqrt{2}}/{2 sqrt{2}} =#
#= {- (1 + sqrt{3}) + i (1 - sqrt{3} - 2 sqrt{2})}/{2 sqrt{2}} ~~#
#~~ -0.97 - 1.26 i#
Tip: in the same way that we can write exponentials as trigonometric functions, we can write trigonometric functions as exponentials.
#cos theta = {e^{i theta} + e^{- i theta}}/2#
#sin theta = {e^{i theta} - e^{-i theta}}/{2i}#
