We have,

\(\displaystyle{{\cos}^{{{2}}}{u}}+{\sin{{u}}}={0}\)

By using the identity, \(\displaystyle{{\cos}^{{{2}}}{u}}={1}−{\sin{{2}}}{u}\) and solving further,

\(\displaystyle{{\cos}^{{{2}}}{u}}+{\sin{{u}}}={0}\)

\(\displaystyle{1}−{{\sin}^{{{2}}}{u}}+{\sin{=}}{0}\)

\(\displaystyle{{\sin}^{{{2}}}{u}}−{\sin{{u}}}−{1}={0}\)

The above equation is quadratic equation in variable sinu.

Using the quadratic equation formula and solving it further, we get our result as

\(\displaystyle{\sin{{u}}}={\frac{{-{\left(-{1}\right)}\pm\sqrt{{{\left(-{1}\right)}^{{{2}}}-{4}{\left({1}\right)}{\left(-{1}\right)}}}}}{{{2}{\left({1}\right)}}}}\)

\(\displaystyle{\sin{{u}}}={\frac{{{1}+\sqrt{{{5}}}}}{{{2}}}},{\frac{{{1}-\sqrt{{{5}}}}}{{{2}}}}\)

Now,

\(\displaystyle{\sin{{u}}}={\frac{{{1}+\sqrt{{{5}}}}}{{{2}}}}\)

No solution occurs because range for sine function lies between -1 to 1

For,

\(\displaystyle{\sin{{u}}}={\frac{{{1}-\sqrt{{{5}}}}}{{{2}}}}\)

\(\displaystyle{\sin{{u}}}={0.618}\)

\(\displaystyle{u}={{\sin}^{{-{1}}}{\left(-{0.618}\right)}}\)

\(\displaystyle{u}=-{{\sin}^{{-{1}}}{\left({0.618}\right)}}\)

\(\displaystyle{u}={38.17}^{{\circ}}\)

\(\displaystyle{u}={360}^{{\circ}}-{38.17}^{{\circ}}\)

\(\displaystyle{u}={321.82}^{{\circ}}\)

Hence,value of u will be \(\displaystyle{321.82}^{{\circ}}\)

\(\displaystyle{{\cos}^{{{2}}}{u}}+{\sin{{u}}}={0}\)

By using the identity, \(\displaystyle{{\cos}^{{{2}}}{u}}={1}−{\sin{{2}}}{u}\) and solving further,

\(\displaystyle{{\cos}^{{{2}}}{u}}+{\sin{{u}}}={0}\)

\(\displaystyle{1}−{{\sin}^{{{2}}}{u}}+{\sin{=}}{0}\)

\(\displaystyle{{\sin}^{{{2}}}{u}}−{\sin{{u}}}−{1}={0}\)

The above equation is quadratic equation in variable sinu.

Using the quadratic equation formula and solving it further, we get our result as

\(\displaystyle{\sin{{u}}}={\frac{{-{\left(-{1}\right)}\pm\sqrt{{{\left(-{1}\right)}^{{{2}}}-{4}{\left({1}\right)}{\left(-{1}\right)}}}}}{{{2}{\left({1}\right)}}}}\)

\(\displaystyle{\sin{{u}}}={\frac{{{1}+\sqrt{{{5}}}}}{{{2}}}},{\frac{{{1}-\sqrt{{{5}}}}}{{{2}}}}\)

Now,

\(\displaystyle{\sin{{u}}}={\frac{{{1}+\sqrt{{{5}}}}}{{{2}}}}\)

No solution occurs because range for sine function lies between -1 to 1

For,

\(\displaystyle{\sin{{u}}}={\frac{{{1}-\sqrt{{{5}}}}}{{{2}}}}\)

\(\displaystyle{\sin{{u}}}={0.618}\)

\(\displaystyle{u}={{\sin}^{{-{1}}}{\left(-{0.618}\right)}}\)

\(\displaystyle{u}=-{{\sin}^{{-{1}}}{\left({0.618}\right)}}\)

\(\displaystyle{u}={38.17}^{{\circ}}\)

\(\displaystyle{u}={360}^{{\circ}}-{38.17}^{{\circ}}\)

\(\displaystyle{u}={321.82}^{{\circ}}\)

Hence,value of u will be \(\displaystyle{321.82}^{{\circ}}\)