Mathematics

This problem asks you to navigate complex logarithms and the tangent function by converting everything to exponential form. The key insight is to recognize that $\tan(i\theta) = i\tanh(\theta)$, and then use the definition of the complex logarithm to simplify $\ln\left(\frac{a-ib}{a+ib}\right)$. Once you express the argument as a pure imaginary number times a real exponent, the hyperbolic tangent formula will naturally produce the real result on the right side.

This problem asks you to navigate complex logarithms and the tangent function by converting everything to exponential form. The key insight is to recognize that $\tan(i\theta) = i\tanh(\theta)$, and then use the definition of the complex logarithm to simplify $\ln\left(\frac{a-ib}{a+ib}\right)$. Once you express the argument as a pure imaginary number times a real exponent, the hyperbolic tangent formula will naturally produce the real result on the right side.