2008/006) If z = cos x + i sin x ,then simplify (z^2 -1)/(z^2 +1)

(z^2-1)/(z^2+ 1) = (z - 1/z)/ (z+ 1/z) ...1

now z = cos x + i sin x ... 2

so z = e^ix

so 1/z = e ^-ix = cos x - i sin x ... 3

add 2 and 3 to get z+ 1/z = 2 cos x

subtract 3 from 2 to get

(z-1/z) = 2i sin x

from 1 by deviding (z^2-1)/z^2+1)= 2i sin x/(2 cos x) = i tan x