Honest answer. I first did it by Excel. The I realized that co-functions of complementary angles are equal, and that cotangent and tangent are reciprocals, and reciprocals have logs which are negatives of each other.
tan 1°= cot 89°=1/tan 89° the log tan 1°=-log tan 89°
Do this for all the pairs 1,89; 2,88; 3,87;... ;45,45
The sum is zero! Sorry about the pairs listing errors. Now corrected.
