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