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