\[~\left( \mathbf{v} \right)~ta{{n}^{2}}~x\text{ }+\text{ }\left( 1\text{ }\text{ }\surd 3 \right)\text{ }tan\text{ }x\text{ }\text{ }\surd 3\text{ }=\text{ }0\]

Or,

\[ta{{n}^{2}}~x\text{ }+\text{ }\left( 1\text{ }\text{ }\surd 3 \right)\text{ }tan\text{ }x\text{ }\text{ }\surd 3\text{ }=\text{ }0\]

\[ta{{n}^{2}}~x\text{ }+\text{ }tan\text{ }x\text{ }\text{ }\surd 3\text{ }tan\text{ }x\text{ }\text{ }\surd 3\text{ }=\text{ }0\]

or,

\[tan\text{ }x\text{ }\left( tan\text{ }x\text{ }+\text{ }1 \right)\text{ }\text{ }\surd 3\text{ }\left( tan\text{ }x\text{ }+\text{ }1 \right)\text{ }=\text{ }0\]

\[\left( tan\text{ }x\text{ }+\text{ }1 \right)\text{ }\left( \text{ }tan\text{ }x\text{ }\text{ }\surd 3 \right)\text{ }=\text{ }0\]

Or,

\[tan\text{ }x\text{ }=\text{ }-1\text{ }or\text{ }tan\text{ }x\text{ }=\text{ }\surd 3\]

As, tan x ϵ (-∞ , ∞) so both values are valid and acceptable.

\[tan\text{ }x\text{ }=\text{ }tan\text{ }\left( -\pi /4 \right)\]

\[or,\text{ }tan\text{ }x\text{ }=\text{ }tan\text{ }\left( \pi /3 \right)\]

\[x\text{ }=\text{ }m\pi \text{ }\text{ }\pi /4\text{ }or\text{ }x\text{ }=\text{ }n\pi \text{ }+\text{ }\pi /3\]

∴ the general solution is

\[x\text{ }=\text{ }m\pi \text{ }\text{ }\pi /4\text{ }or\text{ }n\pi \text{ }+\text{ }\pi /3,\]

where m, n ϵ Z.

\[\left( \mathbf{vi} \right)~3\text{ }co{{s}^{2}}~x\text{ }\text{ }2\surd 3\text{ }sin\text{ }x\text{ }cos\text{ }x\text{ }\text{ }3\text{ }si{{n}^{2}}~x\text{ }=\text{ }0\]

Or,

\[3\text{ }co{{s}^{2}}~x\text{ }\text{ }2\surd 3\text{ }sin\text{ }x\text{ }cos\text{ }x\text{ }\text{ }3\text{ }si{{n}^{2}}~x\text{ }=\text{ }0\]

\[3\text{ }co{{s}^{2}}~x\text{ }\text{ }3\surd 3\text{ }sin\text{ }x\text{ }cos\text{ }x\text{ }+\text{ }\surd 3\text{ }sin\text{ }x\text{ }cos\text{ }x\text{ }\text{ }3\text{ }si{{n}^{2}}~x\text{ }=\text{ }0\]

Or,

\[3\text{ }cos\text{ }x\text{ }\left( cos\text{ }x\text{ }\text{ }\surd 3sin\text{ }x \right)\text{ }+\text{ }\surd 3\text{ }sin\text{ }x\text{ }\left( cos\text{ }x\text{ }\text{ }\surd 3\text{ }sin\text{ }x \right)\text{ }=\text{ }0\]

\[\surd 3\text{ }\left( cos\text{ }x\text{ }\text{ }\surd 3\text{ }sin\text{ }x \right)\text{ }\left( \surd 3\text{ }cos\text{ }x\text{ }+\text{ }sin\text{ }x \right)\text{ }=\text{ }0\]

Or,

\[cos\text{ }x\text{ }\text{ }\surd 3\text{ }sin\text{ }x\text{ }=\text{ }0\text{ }or\text{ }sin\text{ }x\text{ }+\text{ }\surd 3\text{ }cos\text{ }x\text{ }=\text{ }0\]

\[cos\text{ }x\text{ }=\text{ }\surd 3\text{ }sin\text{ }x\text{ }or\text{ }sin\text{ }x\text{ }=\text{ }-\surd 3\text{ }cos\text{ }x\]

or,

\[tan\text{ }x\text{ }=\text{ }1/\surd 3\text{ }or\text{ }tan\text{ }x\text{ }=\text{ }-\surd 3\]

As, tan x ϵ (-∞ , ∞) so both values are valid and acceptable.

\[tan\text{ }x\text{ }=\text{ }tan\text{ }\left( \pi /6 \right)\text{ }or\text{ }tan\text{ }x\text{ }=\text{ }tan\text{ }\left( -\pi /3 \right)\]

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

∴ the general solution is

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

where m, n ϵ Z.