(xi) \[sin\text{ }x\text{ }=\text{ }tan\text{ }x\]

Or,

\[sin\text{ }x\text{ }=\text{ }tan\text{ }x\]

\[sin\text{ }x\text{ }=\text{ }sin\text{ }x/cos\text{ }x\]

or,

\[sin\text{ }x\text{ }cos\text{ }x\text{ }=\text{ }sin\text{ }x\]

\[sin\text{ }x\text{ }\left( cos\text{ }x\text{ }\text{ }1 \right)\text{ }=\text{ }0\]

So,

\[Sin\text{ }x\text{ }=\text{ }0\text{ }or\text{ }cos\text{ }x\text{ }\text{ }1\text{ }=\text{ }0\]

\[Sin\text{ }x\text{ }=\text{ }sin\text{ }0\text{ }\left[ or \right]\text{ }cos\text{ }x\text{ }=\text{ }1\]

Or,

\[Sin\text{ }x\text{ }=\text{ }sin\text{ }0\text{ }\left[ or \right]\text{ }cos\text{ }x\text{ }=\text{ }cos\text{ }0\]

\[x\text{ }=\text{ }n\pi \text{ }\left[ or \right]\text{ }x\text{ }=\text{ }2m\pi \]

∴ the general solution is

\[x\text{ }=\text{ }n\pi \text{ }\left[ or \right]\text{ }2m\pi \]

, where n, m ϵ Z.

\[\left( \mathbf{xii} \right)~sin\text{ }3x\text{ }+\text{ }cos\text{ }2x\text{ }=\text{ }0\]

Or,

\[sin\text{ }3x\text{ }+\text{ }cos\text{ }2x\text{ }=\text{ }0\]

\[cos\text{ }2x\text{ }=\text{ }\text{ }sin\text{ }3x\]

\[cos\text{ }2x\text{ }=\text{ }\text{ }cos\text{ }\left( \pi /2\text{ }\text{ }3x \right)\]

\[\left[ as,\text{ }sin\text{ }A\text{ }=\text{ }cos\text{ }\left( \pi /2\text{ }\text{ }A \right) \right]\]

Or,

\[cos\text{ }2x\text{ }=\text{ }cos\text{ }\left( \pi \text{ }\text{ }\left( \pi /2\text{ }\text{ }3x \right) \right)\]

\[~\left[ as,\text{ }-cos\text{ }A\text{ }=\text{ }cos\text{ }\left( \pi \text{ }\text{ }A \right) \right]\]

\[cos\text{ }2x\text{ }=\text{ }cos\text{ }\left( \pi /2\text{ }+\text{ }3x \right)\] or,

\[2x\text{ }=\text{ }2n\pi \text{ }\pm \text{ }\left( \pi /2\text{ }+\text{ }3x \right)\]

So,

\[2x\text{ }=\text{ }2n\pi \text{ }+\text{ }\left( \pi /2\text{ }+\text{ }3x \right)\]

[or]

\[2x\text{ }=\text{ }2n\pi \text{ }\text{ }\left( \pi /2\text{ }+\text{ }3x \right)\]

\[x\text{ }=\text{ }-\pi /2\text{ }\text{ }2n\pi \text{ }\left[ or \right]\text{ }5x\text{ }=\text{ }2n\pi \text{ }\text{ }\pi /2\]

or,

\[x\text{ }=\text{ }-\pi /2\text{ }\left( 1\text{ }+\text{ }4n \right)\text{ }\left[ or \right]\text{ }x\text{ }=\text{ }\pi /10\text{ }\left( 4n\text{ }\text{ }1 \right)\]

\[x\text{ }=\text{ }\text{ }\pi /2\text{ }\left( 4n\text{ }+\text{ }1 \right)\text{ }\left[ or \right]\text{ }\pi /10\text{ }\left( 4n\text{ }\text{ }1 \right)\]

∴ the general solution is

\[x\text{ }=\text{ }\text{ }\pi /2\text{ }\left( 4n\text{ }+\text{ }1 \right)\text{ }\left[ or \right]\text{ }\pi /10\text{ }\left( 4n\text{ }\text{ }1 \right)\]

\[x\text{ }=\text{ }\pi /2\text{ }\left( 4n\text{ }\text{ }1 \right)\]

\[~\left[ or \right]\text{ }\pi /10\text{ }\left( 4n\text{ }\text{ }1 \right),~~~~~~~~\]

where n ϵ Z