This is nearly equivalent to the integral from 0 to pi/2 of ln(tan(x))) dx.  So that might be a way of solving; and this integral does evaluate to 0.

But notice that the tangent function is symmetric around 45 degrees, so let's look at how tan(45 + t) compares with tan(45 - t)

tan(a+b) = (tan(a) + tan(b))/(1 - tan(a)*tan(b))

tan(45 + t) = (tan(45) + tan(t))/(1 - tan(45)*tan(t))

tan(45 + t) = (1 + tan(t))/(1 - tan(t))

tan(45 - t) = (1 - tan(t))/(1 + tan(t))

So for every value of t, these 2 values are reciprocals.

The log of (a/b) is the opposite of the log of (b/a).

So by pairing (1,89), (2,88), (3,87) ...  

log(tan(1)) + log(tan(89)) = 0

log(tan(2)) + log(tan(88)) = 0

etc

So they all add up to zero.

log(tan(45)) doesn't have a mate to pair up with, but log(1) = 0.

So the answer is zero.