Step 1

\(\displaystyle\sqrt{{2}} \cos{{\left({x}\right)}} \sin{{\left({x}\right)}}+ \cos{{\left({x}\right)}}={0}\)

\(\displaystyle \cos{{\left({x}\right)}}{\left(\sqrt{{2}} \sin{{\left({x}\right)}}+{1}\right)}={0}\)

\(\displaystyle \cos{{\left({x}\right)}}={0}{\quad\text{or}\quad}\sqrt{{2}} \sin{{\left({x}\right)}}+{1}={0}\)

\(\displaystyle \cos{{\left({x}\right)}}={0}{\quad\text{or}\quad} \sin{{\left({x}\right)}}=-\frac{1}{\sqrt{{2}}}\)

first case:

when \(\displaystyle \cos{{x}}={0}\).

\(\displaystyle \cos{{x}}= \cos{{\left(\frac{\pi}{{2}}\right)}}\)

therefore, \(\displaystyle{x}=\frac{\pi}{{2}}+{2}{k}\pi\)

Step 2

when \(\displaystyle \cos{{x}}={0}.\)

\(\displaystyle \cos{{x}}= \cos{{\left(\frac{{{3}\pi}}{{2}}\right)}}\)

therefore, \(\displaystyle{x}=\frac{{{3}\pi}}{{2}}+{2}{k}\pi\)

second case:

when \(\displaystyle \sin{{x}}=\frac{{-{1}}}{\sqrt{{2}}}\)

\(\displaystyle \sin{{x}}= \sin{{\left(\frac{{{5}\pi}}{{4}}\right)}}\)

therefore, \(\displaystyle{x}=\frac{{{5}\pi}}{{4}}+{2}{k}\pi\)

Step 3

when \(\displaystyle \sin{{x}}=\frac{{-{1}}}{\sqrt{{2}}}\)

\(\displaystyle \sin{{x}}= \sin{{\left(\frac{{{7}\pi}}{{4}}\right)}}\)

therefore, \(\displaystyle{x}=\frac{{{7}\pi}}{{4}}+{2}{k}\pi\)

therefore the combined solution is:

\(\displaystyle{x}=\frac{\pi}{{2}}+{2}{k}\pi{\quad\text{or}\quad}{x}=\frac{{{5}\pi}}{{4}}+{2}{k}\pi{\quad\text{or}\quad}{x}=\frac{{{3}\pi}}{{2}}+{2}{k}\pi{\quad\text{or}\quad}{x}=\frac{{{7}\pi}}{{4}}+{2}{k}\pi\)

Step 4

therefore the solution of the given equation \(\displaystyle\sqrt{{2}} \cos{{\left({x}\right)}} \sin{{\left({x}\right)}}+ \cos{{\left({x}\right)}}={0}\) is:

\(\displaystyle{x}=\frac{\pi}{{2}}+{2}{k}\pi{\quad\text{or}\quad}{x}=\frac{{{5}\pi}}{{4}}+{2}{k}\pi{\quad\text{or}\quad}{x}=\frac{{{3}\pi}}{{2}}+{2}{k}\pi{\quad\text{or}\quad}{x}=\frac{{{7}\pi}}{{4}}+{2}{k}\pi\)

\(\displaystyle\sqrt{{2}} \cos{{\left({x}\right)}} \sin{{\left({x}\right)}}+ \cos{{\left({x}\right)}}={0}\)

\(\displaystyle \cos{{\left({x}\right)}}{\left(\sqrt{{2}} \sin{{\left({x}\right)}}+{1}\right)}={0}\)

\(\displaystyle \cos{{\left({x}\right)}}={0}{\quad\text{or}\quad}\sqrt{{2}} \sin{{\left({x}\right)}}+{1}={0}\)

\(\displaystyle \cos{{\left({x}\right)}}={0}{\quad\text{or}\quad} \sin{{\left({x}\right)}}=-\frac{1}{\sqrt{{2}}}\)

first case:

when \(\displaystyle \cos{{x}}={0}\).

\(\displaystyle \cos{{x}}= \cos{{\left(\frac{\pi}{{2}}\right)}}\)

therefore, \(\displaystyle{x}=\frac{\pi}{{2}}+{2}{k}\pi\)

Step 2

when \(\displaystyle \cos{{x}}={0}.\)

\(\displaystyle \cos{{x}}= \cos{{\left(\frac{{{3}\pi}}{{2}}\right)}}\)

therefore, \(\displaystyle{x}=\frac{{{3}\pi}}{{2}}+{2}{k}\pi\)

second case:

when \(\displaystyle \sin{{x}}=\frac{{-{1}}}{\sqrt{{2}}}\)

\(\displaystyle \sin{{x}}= \sin{{\left(\frac{{{5}\pi}}{{4}}\right)}}\)

therefore, \(\displaystyle{x}=\frac{{{5}\pi}}{{4}}+{2}{k}\pi\)

Step 3

when \(\displaystyle \sin{{x}}=\frac{{-{1}}}{\sqrt{{2}}}\)

\(\displaystyle \sin{{x}}= \sin{{\left(\frac{{{7}\pi}}{{4}}\right)}}\)

therefore, \(\displaystyle{x}=\frac{{{7}\pi}}{{4}}+{2}{k}\pi\)

therefore the combined solution is:

\(\displaystyle{x}=\frac{\pi}{{2}}+{2}{k}\pi{\quad\text{or}\quad}{x}=\frac{{{5}\pi}}{{4}}+{2}{k}\pi{\quad\text{or}\quad}{x}=\frac{{{3}\pi}}{{2}}+{2}{k}\pi{\quad\text{or}\quad}{x}=\frac{{{7}\pi}}{{4}}+{2}{k}\pi\)

Step 4

therefore the solution of the given equation \(\displaystyle\sqrt{{2}} \cos{{\left({x}\right)}} \sin{{\left({x}\right)}}+ \cos{{\left({x}\right)}}={0}\) is:

\(\displaystyle{x}=\frac{\pi}{{2}}+{2}{k}\pi{\quad\text{or}\quad}{x}=\frac{{{5}\pi}}{{4}}+{2}{k}\pi{\quad\text{or}\quad}{x}=\frac{{{3}\pi}}{{2}}+{2}{k}\pi{\quad\text{or}\quad}{x}=\frac{{{7}\pi}}{{4}}+{2}{k}\pi\)