On Ramanujan type identities and Cardano formula

Abstract

In this paper we will prove some Ramanujan type identities such as \begin{align*} &\sqrt[3]{\sin\left(\frac{\pi}{9}\right)} + \sqrt[3]{\sin\left(\frac{2\pi}{9}\right)} + \sqrt[3]{\sin\left(\frac{14\pi}{9}\right)} \\ & = \left(-\frac{\sqrt[18]{3}}{2}\right) \left(\sqrt[3]{6+3\left(\sqrt[3]{6-3\sqrt[3]{9}}+ \sqrt[3]{3-3\sqrt[3]{9}}\right)}\right), \end{align*} \begin{align*} &\sqrt[3]{\tan\left(\frac{\pi}{9}\right)} + \sqrt[3]{\tan\left(\frac{4\pi}{9}\right)} + \sqrt[3]{\tan\left(\frac{7\pi}{9}\right)} \\ & = \left(-\sqrt[18]{3}\right) \left(\sqrt[3]{-3\sqrt[3]{3}+6+3(\sqrt[3]{\!21 - 3(3\sqrt[3]{3}\!-\!\sqrt[3]{9}) } - \sqrt[3]{\!3 + 3(3\sqrt[3]{3}\!+\!\sqrt[3]{9})})}\right). \end{align*}