Question

If $\alpha=\log _{ e } \sqrt{\cot \frac{\pi}{12}}$, then find the value of $\left(\frac{\displaystyle\sum_{ k =0}^{\infty} e ^{-2 k \alpha}}{\displaystyle\sum_{ k =0}^{\infty}(-1)^{ k } e ^{-2 k \alpha}}\right)^{2}$

Answer 1

Expression
$\frac{1+ e ^{-2 \alpha}+ e ^{-4 \alpha}+\ldots \infty}{1- e ^{-2 \alpha}+ e ^{-4 \alpha}+\ldots \infty}$
$=\frac{1+ e ^{-2 \alpha}}{1- e ^{-2 \alpha}}=\left(\frac{ e ^{2 \alpha}+1}{ e ^{2 \alpha}-1}\right)$
As, $e ^{\alpha}=\sqrt{\cot \frac{\pi}{12}} $
$\Rightarrow e ^{2 \alpha}=(2+\sqrt{3})$
$\therefore d$ Expression $=\sqrt{3}$