\[~\left( \mathbf{iii} \right)~cosec\text{ }x\text{ }=\text{ }\text{ }\surd 2\]

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.