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