\[~\left( \mathbf{iii} \right)~tan\text{ }3x\text{ }+\text{ }tan\text{ }x\text{ }=\text{ }2\text{ }tan\text{ }2x\]
Or,
\[tan\text{ }3x\text{ }+\text{ }tan\text{ }x\text{ }=\text{ }2\text{ }tan\text{ }2x\]
\[tan\text{ }3x\text{ }+\text{ }tan\text{ }x\text{ }=\text{ }tan\text{ }2x\text{ }+\text{ }tan\text{ }2x\]
Or,
\[tan\text{ }3x\text{ }\text{ }tan\text{ }2x\text{ }=\text{ }tan\text{ }2x\text{ }\text{ }tan\text{ }x\]
By using the formula,
\[tan\text{ }\left( A-B \right)\text{ }=\text{ }\left[ tan\text{ }A\text{ }\text{ }tan\text{ }B \right]\text{ }/\text{ }\left[ 1\text{ }+\text{ }tan\text{ }A\text{ }tan\text{ }B \right]\]
so,
\[\left[ \left( tan\text{ }3x\text{ }\text{ }tan\text{ }2x \right)\text{ }\left( 1+tan\text{ }3x\text{ }tan\text{ }2x \right) \right]\text{ }/\text{ }\left[ 1\text{ }+\text{ }tan\text{ }3x\text{ }tan\text{ }2x \right]\]
\[~=\text{ }\left[ \left( tan\text{ }2x-tan\text{ }x \right)\text{ }\left( 1+tan\text{ }x\text{ }tan\text{ }2x \right) \right]\text{ }/\text{ }\left[ 1\text{ }+\text{ }tan\text{ }2x\text{ }tan\text{ }x \right]\]
Or,
\[tan\text{ }\left( 3x\text{ }\text{ }2x \right)\text{ }\left( 1\text{ }+\text{ }tan\text{ }3x\text{ }tan\text{ }2x \right)\]
\[=\text{ }tan\text{ }\left( 2x\text{ }\text{ }x \right)\text{ }\left( 1\text{ }+\text{ }tan\text{ }x\text{ }tan\text{ }2x \right)\]
\[tan\text{ }x\text{ }\left[ 1\text{ }+\text{ }tan\text{ }3x\text{ }tan\text{ }2x\text{ }\text{ }1\text{ }\text{ }tan\text{ }2x\text{ }tan\text{ }x \right]\text{ }=\text{ }0\]
or,
\[tan\text{ }x\text{ }tan\text{ }2x\text{ }\left( tan\text{ }3x\text{ }\text{ }tan\text{ }x \right)\text{ }=\text{ }0\]
so,
\[tan\text{ }x\text{ }=\text{ }0\text{ }or\text{ }tan\text{ }2x\text{ }=\text{ }0\]
or
\[\left( tan\text{ }3x\text{ }\text{ }tan\text{ }x \right)\text{ }=\text{ }0\]
\[tan\text{ }x\text{ }=\text{ }0\]
or
\[tan\text{ }2x\text{ }=\text{ }0\]
or
\[tan\text{ }3x\text{ }=\text{ }tan\text{ }x\]
\[x\text{ }=\text{ }n\pi \]
or
\[2x\text{ }=\text{ }m\pi \]
or
\[3x\text{ }=\text{ }k\pi \text{ }+\text{ }x\]
\[x\text{ }=\text{ }n\pi \]
or
\[x\text{ }=\text{ }m\pi /2\]
or
\[2x\text{ }=\text{ }k\pi ,\]
\[x\text{ }=\text{ }n\pi \]
or
\[x\text{ }=\text{ }m\pi /2\]
or
\[x\text{ }=\text{ }k\pi /2\]
∴ the general solution is
\[x\text{ }=\text{ }n\pi \text{ }or\text{ }m\pi /2\text{ }or\text{ }k\pi /2,\]
where, m, n, k ∈ Z.