# Approximation for fractional function for given boundary conditions

I try to find an approximation for the following expression

$$\tan\phi(x)=\frac{\mathrm{si}(x)\sin(kx)-\mathrm{si}(x-\tau)\sin(k(x-\tau))}{\mathrm{si}(x)\cos(kx)-\mathrm{si}(x-\tau)\cos(k(x-\tau))} \tag1$$

in which $k\gg\tau$ and $\tau\ll 1$.

The assumption $\tau\ll1$ means $\mathrm{si}(x)\approx \mathrm{si}(x -\tau)$ leads to alternate form according to wolframalpha

$$\tan\phi(x)\approx\frac{\mathrm{si}(x)\sin(kx)-\mathrm{si}(x)\sin(k(x-\tau))}{\mathrm{si}(x)\cos(kx)-\mathrm{si}(x)\cos(k(x-\tau))}=-\cot(k(x-\frac{\tau}{2}))\tag2$$

At this point I do not know if I make a proper approximation or a huge mistake. Is there a another way to approximate the first expression or do I approximate in a right way?