\[~\left( \mathbf{iii} \right)~cosec\text{ }x\text{ }=\text{ }\text{ }\surd 2\]
Or,
\[1/sin\text{ }x\text{ }=\text{ }\text{ }\surd 2\]
\[~\left[ as,\text{ }cosec\text{ }x\text{ }=\text{ }1/sin\text{ }x \right]\]
\[Sin\text{ }x\text{ }=\text{ }-1/\surd 2\]
\[=\text{ }sin\text{ }\left[ \pi \text{ }+\text{ }\pi /4 \right]\]
\[=\text{ }sin\text{ }5\pi /4\text{ }or\text{ }sin\text{ }\left( -\pi /4 \right)\]
∴ the general solution is
\[x\text{ }=\text{ }n\pi \text{ }+\text{ }{{\left( -1 \right)}^{n+1}}~\pi /4\] ,
where n ϵ Z.
\[\left( \mathbf{iv} \right)~sec\text{ }x\text{ }=\text{ }\surd 2\]
or,
\[1/cos\text{ }x\text{ }=\text{ }\surd 2\]
\[~\left[ as,\text{ }sec\text{ }x\text{ }=\text{ }1/cos\text{ }x \right]\]
\[Cos\text{ }x\text{ }=\text{ }1/\surd 2\]
\[=\text{ }cos\text{ }\pi /4\]
∴ the general solution is
\[x\text{ }=\text{ }2n\pi ~\pm \text{ }\pi /4,\]
where n ϵ Z.