\[~\left( \mathbf{iii} \right)~sin\text{ }9x\text{ }=\text{ }sin\text{ }x\]

Or,

\[Sin\text{ }9x\text{ }\text{ }sin\text{ }x\text{ }=\text{ }0\]

Using transformation formula,

\[Sin\text{ }A\text{ }\text{ }sin\text{ }B\text{ }=\text{ }2\text{ }cos\text{ }\left( A+B \right)/2\text{ }sin\text{ }\left( A-B \right)/2\]

So,

\[=\text{ }2\text{ }cos\text{ }\left( 9x+x \right)/2\text{ }sin\text{ }\left( 9x-x \right)/2\]

\[=>\text{ }cos\text{ }5x\text{ }sin\text{ }4x\text{ }=\text{ }0\]

\[Cos\text{ }5x\text{ }=\text{ }0\text{ }or\text{ }sin\text{ }4x\text{ }=\text{ }0\]

Let us verify both the expressions,

\[Cos\text{ }5x\text{ }=\text{ }0\]

\[Cos\text{ }5x\text{ }=\text{ }cos\text{ }\pi /2\]

\[5x\text{ }=\text{ }\left( 2n\text{ }+\text{ }1 \right)\pi /2\]

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

where n ϵ Z.

\[sin\text{ }4x\text{ }=\text{ }0\]

\[sin\text{ }4x\text{ }=\text{ }sin\text{ }0\]

\[sin\text{ }4x\text{ }=\text{ }sin\text{ }0\]

\[4x\text{ }=\text{ }n\pi \]

\[x\text{ }=\text{ }n\pi /4,\]

where n ϵ Z.

∴ the general solution is

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

where n ϵ Z.

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

Or,

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

or,

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

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

\[\pi /2\text{ }\text{ }2x\text{ }=\text{ }2n\pi \text{ }\pm \text{ }3x\]

or,

\[\pi /2\text{ }\text{ }2x\text{ }=\text{ }2n\pi \text{ }+\text{ }3x\]

[or]

\[\pi /2\text{ }\text{ }2x\text{ }=\text{ }2n\pi \text{ }\text{ }3x\pi /2\text{ }\text{ }2x\text{ }=\text{ }2n\pi \text{ }\text{ }3x\]

\[5x\text{ }=\text{ }\pi /2\text{ }+\text{ }2n\pi \]

[or]

\[x\text{ }=\text{ }2n\pi \text{ }\text{ }\pi /2\]

Or,

\[5x\text{ }=\text{ }\pi /2\text{ }\left( 1\text{ }+\text{ }4n \right)\]

[or]

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

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

[or]

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

or,

∴ the general solution is

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

or,

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

where n ϵ Z.