The general solution of any trigonometric equation is given as:
\[sin\text{ }x\text{ }=\text{ }sin\text{ }y,\]
or,
\[x\text{ }=\text{ }n\pi \text{ }+\text{ }{{\left( \text{ }1 \right)}^{n~}}y,\]
where n ∈ Z.
or,
\[x\text{ }=\text{ }2n\pi ~\pm ~y,\]
where n ∈ Z.
\[tan\text{ }x\text{ }=\text{ }tan\text{ }y,\]
\[x\text{ }=\text{ }n\pi ~+\text{ }y,\]
where n ∈ Z.
\[\left( \mathbf{i} \right)~cos\text{ }x\text{ }+\text{ }cos\text{ }2x\text{ }+\text{ }cos\text{ }3x\text{ }=\text{ }0\]
Or,
\[cos\text{ }x\text{ }+\text{ }cos\text{ }2x\text{ }+\text{ }cos\text{ }3x\text{ }=\text{ }0\]
using transformation formula we get,
\[cos\text{ }2x\text{ }+\text{ }\left( cos\text{ }x\text{ }+\text{ }cos\text{ }3x \right)\text{ }=\text{ }0\]
by using the formula,
\[cos\text{ }A\text{ }+\text{ }cos\text{ }B\text{ }=\text{ }2\text{ }cos\text{ }\left( A+B \right)/2\text{ }cos\text{ }\left( A-B \right)/2\]
\[cos\text{ }2x\text{ }+\text{ }2\text{ }cos\text{ }\left( 3x+x \right)/2\text{ }cos\text{ }\left( 3x-x \right)/2\text{ }=\text{ }0\]
\[cos\text{ }2x\text{ }+\text{ }2cos\text{ }2x\text{ }cos\text{ }x\text{ }=\text{ }0\]
or,
\[cos\text{ }2x\text{ }\left( \text{ }1\text{ }+\text{ }2\text{ }cos\text{ }x \right)\text{ }=\text{ }0\]
\[cos\text{ }2x\text{ }=\text{ }0\text{ }or\text{ }1\text{ }+\text{ }2cos\text{ }x\text{ }=\text{ }0\]
or,
\[cos\text{ }2x\text{ }=\text{ }cos\text{ }0\text{ }or\text{ }cos\text{ }x\text{ }=\text{ }-1/2\]
or
\[cos\text{ }x\text{ }=\text{ }cos\text{ }\left( \pi \text{ }\text{ }\pi /3 \right)\]
or,
\[cos\text{ }2x\text{ }=\text{ }cos\text{ }\pi /2\]
\[~or\text{ }cos\text{ }x\text{ }=\text{ }cos\text{ }\left( 2\pi /3 \right)\]
\[2x\text{ }=\text{ }\left( 2n\text{ }+\text{ }1 \right)\text{ }\pi /2\]
Or
\[x\text{ }=\text{ }2m\pi \text{ }\pm \text{ }2\pi /3\]
\[x\text{ }=\text{ }\left( 2n\text{ }+\text{ }1 \right)\text{ }\pi /4\]
Or
\[~x\text{ }=\text{ }2m\pi \text{ }\pm \text{ }2\pi /3\]
∴ the general solution is
\[x\text{ }=\text{ }\left( 2n\text{ }+\text{ }1 \right)\text{ }\pi /4\]
or
\[2m\pi \text{ }\pm \text{ }2\pi /3,\]
where m, n ϵ Z.
\[\left( \mathbf{ii} \right)~cos\text{ }x\text{ }+\text{ }cos\text{ }3x\text{ }\text{ }cos\text{ }2x\text{ }=\text{ }0\]
Or,
\[cos\text{ }x\text{ }+\text{ }cos\text{ }3x\text{ }\text{ }cos\text{ }2x\text{ }=\text{ }0\]
using transformation formula,
\[cos\text{ }x\text{ }\text{ }cos\text{ }2x\text{ }+\text{ }cos\text{ }3x\text{ }=\text{ }0\]
\[\text{ }cos\text{ }2x\text{ }+\text{ }\left( cos\text{ }x\text{ }+\text{ }cos\text{ }3x \right)\text{ }=\text{ }0\]
By using the formula,
\[cos\text{ }A\text{ }+\text{ }cos\text{ }B\text{ }=\text{ }2\text{ }cos\text{ }\left( A+B \right)/2\text{ }cos\text{ }\left( A-B \right)/2\]
\[\text{ }cos\text{ }2x\text{ }+\text{ }2\text{ }cos\text{ }\left( 3x+x \right)/2\text{ }cos\text{ }\left( 3x-x \right)/2\text{ }=\text{ }0\]
\[\text{ }cos\text{ }2x\text{ }+\text{ }2cos\text{ }2x\text{ }cos\text{ }x\text{ }=\text{ }0\]
\[cos\text{ }2x\text{ }\left( \text{ }-1\text{ }+\text{ }2\text{ }cos\text{ }x \right)\text{ }=\text{ }0\]
or,
\[cos\text{ }2x\text{ }=\text{ }0\text{ }or\text{ }-1\text{ }+\text{ }2cos\text{ }x\text{ }=\text{ }0\]
\[cos\text{ }2x\text{ }=\text{ }cos\text{ }0\]
or,
\[~cos\text{ }x\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\]
\[cos\text{ }2x\text{ }=\text{ }cos\text{ }\pi /2\]
or
\[cos\text{ }x\text{ }=\text{ }cos\text{ }\left( \pi /3 \right)\]
\[2x\text{ }=\text{ }\left( 2n\text{ }+\text{ }1 \right)\text{ }\pi /2\]
Or
\[x\text{ }=\text{ }\left( 2n\text{ }+\text{ }1 \right)\text{ }\pi /4\]
or
\[~x\text{ }=\text{ }2m\pi \text{ }\pm \text{ }\pi /3\]
∴ the general solution is
\[x\text{ }=\text{ }\left( 2n\text{ }+\text{ }1 \right)\text{ }\pi /4\text{ }or\text{ }2m\pi \text{ }\pm \text{ }\pi /3,\]
where m, n ϵ Z.