\[\left( \mathbf{vii} \right)~tan\text{ }2x\text{ }tan\text{ }x\text{ }=\text{ }1\]

Or,

\[tan\text{ }2x\text{ }tan\text{ }x\text{ }=\text{ }1\]

\[tan\text{ }2x\text{ }=\text{ }1/tan\text{ }x\]

\[=\text{ }cot\text{ }x\]

\[tan\text{ }2x\text{ }=\text{ }tan\text{ }\left( \pi /2\text{ }\text{ }x \right)\]

\[~\left[ as,\text{ }cot\text{ }A\text{ }=\text{ }tan\text{ }\left( \pi /2\text{ }\text{ }A \right) \right]\]

\[2x\text{ }=\text{ }n\pi \text{ }+\text{ }\pi /2\text{ }\text{ }x\]

\[3x\text{ }=\text{ }n\pi \text{ }+\text{ }\pi /2\]

Or,

\[x\text{ }=\text{ }n\pi /3\text{ }+\text{ }\pi /6\]

∴ the general solution is

\[x\text{ }=\text{ }n\pi /3\text{ }+\text{ }\pi /6,\]

where n ϵ Z.

\[\left( \mathbf{viii} \right)~tan\text{ }mx\text{ }+\text{ }cot\text{ }nx\text{ }=\text{ }0\]

Or,

\[tan\text{ }mx\text{ }+\text{ }cot\text{ }nx\text{ }=\text{ }0\]

\[tan\text{ }mx\text{ }=\text{ }\text{ }cot\text{ }nx\]

or,

\[=\text{ }\text{ }tan\text{ }\left( \pi /2\text{ }\text{ }nx \right)\]

\[\left[ as,\text{ }cot\text{ }A\text{ }=\text{ }tan\text{ }\left( \pi /2\text{ }\text{ }A \right) \right]\]

\[tan\text{ }mx\text{ }=\text{ }tan\text{ }\left( nx\text{ }+\text{ }\pi /2 \right)\]

\[\left[ as,\text{ }\text{ }tan\text{ }A\text{ }=\text{ }tan\text{ }-A \right]~\]

\[mx\text{ }=\text{ }k\pi \text{ }+\text{ }nx\text{ }+\text{ }\pi /2\]

or,

\[\left( m\text{ }\text{ }n \right)\text{ }x\text{ }=\text{ }k\pi \text{ }+\text{ }\pi /2\]

\[\left( m\text{ }\text{ }n \right)\text{ }x\text{ }=\text{ }\pi \text{ }\left( 2k\text{ }+\text{ }1 \right)/2\]

Or,

\[x\text{ }=\text{ }\pi \text{ }\left( 2k\text{ }+\text{ }1 \right)/2\left( m\text{ }\text{ }n \right)\]

∴ the general solution is

\[x\text{ }=\text{ }\pi \text{ }\left( 2k\text{ }+\text{ }1 \right)/2\left( m\text{ }\text{ }n \right),\]

where m, n, k ϵ Z.