f(x) = ln(cos(x)
The domain is when cos(x) is a positive value.
(cos(x)) > 0
We know that:
cos0 = 1
cos pi/2 = 0
cos pi = -1
cos 3pi/2 = 0
cos 2pi= 1
Then cos(x) values are between -1, and 1
in order for the function to be defined, then the domain is when cos(x) > 0 which are the interval:
(0, pi/2) ( 3pi/2, 2pi)
Then the domain is :
x = (0, pi/2) U (3pi/2, 2pi)
f(x) = ln(cosx)
Here f(x) is real when the logarithm is of positive values.
Therefore cosx > 0 for ln(cosx) to exist.
cosx > 0 when x is in ithe interval : 90 degree <=x =< 90 degree or pi/2 radian <=x=< pi/2 radian . This could be expressed in the interval notation as: x in the interval (-90 deg , 90 deg) or x is in the interval (-pi/2 radian , pi/2 radian).