\[\left( \mathbf{ix} \right)~tan\text{ }px\text{ }=\text{ }cot\text{ }qx\]
Or,
\[tan\text{ }px\text{ }=\text{ }cot\text{ }qx\]
or,
\[tan\text{ }px\text{ }=\text{ }tan\text{ }\left( \pi /2\text{ }\text{ }qx \right)\]
\[~\left[ as,\text{ }cot\text{ }A\text{ }=\text{ }tan\text{ }\left( \pi /2\text{ }\text{ }A \right) \right]\]
\[px\text{ }=\text{ }n\pi \text{ }\pm \text{ }\left( \pi /2\text{ }\text{ }qx \right)\]
\[\left( p\text{ }+\text{ }q \right)\text{ }x\text{ }=\text{ }n\pi \text{ }+\text{ }\pi /2\]
\[x\text{ }=\text{ }n\pi /\left( p+q \right)\text{ }+\text{ }\pi /2\left( p+q \right)\]
\[=\text{ }\pi \text{ }\left( 2n\text{ }+1 \right)/\text{ }2\left( p+q \right)\]
∴ the general solution is
\[x\text{ }=\text{ }\pi \text{ }\left( 2n\text{ }+1 \right)/\text{ }2\left( p+q \right),\]
where n ϵ Z.
\[\left( \mathbf{x} \right)~sin\text{ }2x\text{ }+\text{ }cos\text{ }x\text{ }=\text{ }0\]
Or,
\[sin\text{ }2x\text{ }+\text{ }cos\text{ }x\text{ }=\text{ }0\]
\[cos\text{ }x\text{ }=\text{ }\text{ }sin\text{ }2x\]
Or,
\[cos\text{ }x\text{ }=\text{ }\text{ }cos\text{ }\left( \pi /2\text{ }\text{ }2x \right)\]
\[~\left[ as,\text{ }sin\text{ }A\text{ }=\text{ }cos\text{ }\left( \pi /2\text{ }\text{ }A \right) \right]\]
\[=\text{ }cos\text{ }\left( \pi \text{ }\text{ }\left( \pi /2\text{ }\text{ }2x \right) \right)\]
\[~\left[ as,\text{ }-cos\text{ }A\text{ }=\text{ }cos\text{ }\left( \pi \text{ }\text{ }A \right) \right]\]
\[=\text{ }cos\text{ }\left( \pi /2\text{ }+\text{ }2x \right)\]
\[x\text{ }=\text{ }2n\pi \text{ }\pm \text{ }\left( \pi /2\text{ }+\text{ }2x \right)\]
So,
\[x\text{ }=\text{ }2n\pi \text{ }+\text{ }\left( \pi /2\text{ }+\text{ }2x \right)\]
[or]
\[x\text{ }=\text{ }2n\pi \text{ }\text{ }\left( \pi /2\text{ }+\text{ }2x \right)\]
or,
\[x\text{ }=\text{ }\text{ }\pi /2\text{ }\text{ }2n\pi \]
[or]
\[3x\text{ }=\text{ }2n\pi \text{ }\text{ }\pi /2\]
\[x\text{ }=\text{ }\text{ }\pi /2\text{ }\left( 1\text{ }+\text{ }4n \right)\]
[or]
\[x\text{ }=\text{ }\pi /6\text{ }\left( 4n\text{ }\text{ }1 \right)\]
∴ the general solution is
\[x\text{ }=\text{ }\text{ }\pi /2\text{ }\left( 1\text{ }+\text{ }4n \right),\]
where n ϵ Z.
\[~x\text{ }=\text{ }\pi /6\text{ }\left( 4n\text{ }\text{ }1 \right)\]
\[x\text{ }=\text{ }\pi /2\text{ }\left( 4n\text{ }\text{ }1 \right),\]
where n ϵ Z.
\[~x\text{ }=\text{ }\pi /6\text{ }\left( 4n\text{ }\text{ }1 \right),\]
where n ϵ Z.