What is the most simplified formula for this fraction?

Question

Let's consider the fraction:

$$ \frac{1-\mathrm{e}^{-T/4}}{1-\mathrm{e}^{-T}}. $$

What is the most simplified expression of the above expression, I tried factorizing with the half-angle, but I come up with this complicated expression:

$$ \mathrm{e}^{3T/8}\frac{\sinh(T/8)}{\sinh(T/2)}. $$

Answer 1

Put $$e^{-\frac T4}=x$$

the fraction becomes

$$\frac{1-x}{1-x^4}=\frac{1}{1+x+x^2+x^3}$$ $$=\frac{1}{(1+x)(1+x^2)}$$

Answer 2

$$ \frac{1-\mathrm{e}^{-\frac{T}{4}}}{1-\mathrm{e}^{-T}}\cdot \frac{\mathrm{e}^{\frac{T}{2}}}{\mathrm{e}^{\frac{T}{2}}}=\frac{\mathrm{e}^{\frac{T}{2}}-\mathrm{e}^{\frac{T}{4}}}{\mathrm{e}^{\frac{T}{2}}-\mathrm{e}^{-\frac{T}{2}}}=\frac{2}{\mathrm{e}^{\frac{T}{2}}-\mathrm{e}^{-\frac{T}{2}}}\cdot\frac{\mathrm{e}^{\frac{T}{2}}-\mathrm{e}^{\frac{T}{4}}}{2}=\frac{1}{\sinh \frac{T}{2}}\cdot \mathrm{e}^{\frac{T}{4}} \cdot\frac{\mathrm{e}^{\frac{T}{4}}-1}{2}\cdot \frac{\mathrm{e}^{\frac{T}{8}}}{\mathrm{e}^{\frac{T}{8}}}=\frac{1}{\sinh \frac{T}{2}}\cdot\frac{\mathrm{e}^{\frac{T}{8}}-\mathrm{e}^{-\frac{T}{8}}}{2}\cdot\frac{\mathrm{e}^{\frac{T}{4}}}{\mathrm{e}^{\frac{T}{8}}}=\mathrm{e}^{3T/8}\frac{\sinh(T/8)}{\sinh(T/2)}$$