$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 \over 2}}
\newcommand{\ic}{\mathrm{i}}
\newcommand{\iff}{\Leftrightarrow}
\newcommand{\imp}{\Longrightarrow}
\newcommand{\ol}[1]{\overline{#1}}
\newcommand{\pars}[1]{\left(\, #1 \,\right)}
\newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
\newcommand{\root}[2][]{\,\sqrt[#1]{\, #2 \,}\,}
\newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
\newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$
\begin{align}
&\color{#f00}{\sin\pars{x} - \sin\pars{3x} + \sin\pars{5x}}
\\[3mm] = &\
\Im\sum_{k = 0}^{2}\pars{-1}^{k}\expo{\ic\pars{2k + 1}x} =
\Im\bracks{\expo{\ic x}\,
{\pars{-\expo{2\ic x}}^{3} - 1 \over -\expo{2\ic x} - 1}} =
\sin\pars{3x}\,{\cos\pars{3x} \over \color{#f00}{\cos\pars{x}}}\tag{1}
\\
&\ --------------------------------
\\
&\color{#f00}{\cos\pars{x} + \cos\pars{3x} + \cos\pars{5x}}
\\[3mm] = &\
\Re\sum_{k = 0}^{2}\expo{\ic\pars{2k + 1}x} =
\Re\bracks{\expo{\ic x}\,
{\pars{\expo{2\ic x}}^{3} - 1 \over \expo{2\ic x} - 1}} =
\cos\pars{3x}\,{\sin\pars{3x} \over \color{#f00}{\sin\pars{x}}}\tag{2}
\\
&\ --------------------------------
\\ &\
\mbox{Then,}\quad {\pars{1} \over \pars{2}} = \color{#f00}{\tan\pars{x}}
\end{align}