Consider the operator $\frac D{e^D-1}$ which we will call "shadow":
$$\frac {D}{e^D-1}f(x)=\frac1{2 \pi }\int_{-\infty }^{+\infty } \frac{e^{-iwx}}{e^{-i w}-1}\int_{-\infty }^{+\infty } e^{i t w} f'(t) \, dt \, dw$$
The integrals here should be understood as Fourier transforms.
Now, intuitively, why the following?
$$\frac {D_x}{e^{D_x}-1} \left[\frac1\pi\ln \left(\frac{x+1/2 +\frac{z}{\pi }}{x+1/2 -\frac{z}{\pi }}\right)\right]|_{x=0}=\tan z$$
There are other examples where shadow converts trigonometric functions into inverse trigonometric, logarithms to exponents, etc:
$$\frac {D_x}{e^{D_x}-1} \left[\frac1{\pi }\ln \left(\frac{x+1-\frac{z}{\pi }}{x+\frac{z}{\pi }}\right)\right]|_{x=0}=\cot z$$