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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:

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

Xander Henderson
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Victor
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    I have voted to close this question as "primarily opinion based." The notion of what is "most simple" is not an objective mathematical notation, and what is simple in one context may be considered complicated in another. For example, elementary students are often told to simplify $1/\sqrt{2}$ to $\sqrt{2}/2$, though I would argue that the former is "more simple" than the latter for a lot of purposes. In the case of this question, the original expression seems pretty simple to me. – Xander Henderson Sep 01 '19 at 20:26

2 Answers2

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

the fraction becomes

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

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

tommaso faustini
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