Question 178
Evaluate the sum
$$
\frac{1}{1+\tan 1^{\circ}}+\frac{1}{1+\tan 2^{\circ}}+\frac{1}{1+\tan 3^{\circ}}+\cdots+\frac{1}{1+\tan 89^{\circ}}
$$
(The tangent $(\tan)$ of an angle $\alpha$ is the ratio $BC/AC$ in a right triangle $ABC$ with $\angle C=90^{\circ}$ and $\angle A=\alpha$, and its value does not depend on the triangle used.)
$$
\frac{1}{1+\tan 1^{\circ}}+\frac{1}{1+\tan 2^{\circ}}+\frac{1}{1+\tan 3^{\circ}}+\cdots+\frac{1}{1+\tan 89^{\circ}}
$$
(The tangent $(\tan)$ of an angle $\alpha$ is the ratio $BC/AC$ in a right triangle $ABC$ with $\angle C=90^{\circ}$ and $\angle A=\alpha$, and its value does not depend on the triangle used.)