find the domain of tan2x/(6cosx+2sin2x)

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$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=\frac{\tan \left(2\mathrm{x}\right)}{6\mathrm{cosx}+2\sin 2\mathrm{x}} \\ \tan \mathrm{is} \mathrm{undefined} \mathrm{at} \mathrm{odd} \mathrm{intgeral} \mathrm{multiple} \mathrm{of} \frac{\mathrm{\pi }}{2} \\ \Rightarrow 2\mathrm{x}\ne \frac{\left(2\mathrm{n}+1\right)\mathrm{\pi }}{2} \\ \Rightarrow \mathrm{x}\ne \frac{\left(2\mathrm{n}+1\right)\mathrm{\pi }}{4} \\ \mathrm{Also} \mathrm{denominator} \mathrm{must} \mathrm{not} \mathrm{be} 0, \\ \Rightarrow 6\mathrm{cosx}+2\sin 2\mathrm{x}\ne 0 \\ \Rightarrow 6\mathrm{cosx}+2\left(2\mathrm{sinx}.\mathrm{cosx}\right)\ne 0 \\ \Rightarrow 2\mathrm{cosx}\left(3+2\mathrm{sinx}\right)\ne 0 \\ \Rightarrow \mathrm{cosx}\ne 0 \mathrm{or} 3+2\mathrm{sinx}\ne 0 \\ \Rightarrow \mathrm{cosx}\ne 0 \mathrm{or} \mathrm{sinx}\ne -\frac{3}{2} \left[\mathrm{Minimum} \mathrm{value} \mathrm{of} \mathrm{sinx} \mathrm{is} -01\right] \\ \mathrm{Hence} \\ \mathrm{cosx}\ne 0 \\ \cos \mathrm{is} 0 \mathrm{at} \mathrm{odd} \mathrm{intgeral} \mathrm{multiple} \mathrm{of} \frac{\mathrm{\pi }}{2} \\ \Rightarrow \mathrm{x}\ne \frac{\left(2\mathrm{n}+1\right)\mathrm{\pi }}{2} \\ \mathbf{Hence}\mathbf{ }\mathbf{domain}\mathbf{ }\mathbf{is}\mathbf{ }\left(\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\right)\mathbf{-}\left\{\frac{\left(\mathbf{2}\mathbf{n}\mathbf{+}\mathbf{1}\right)\mathbf{\pi }}{\mathbf{4}}\mathbf{,}\frac{\left(\mathbf{2}\mathbf{n}\mathbf{+}\mathbf{1}\right)\mathbf{\pi }}{\mathbf{2}}\right\} \end{gathered}$$

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