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


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?


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