Notes: `cos(120 + 45), cos(120) + cos(45)` Find the exact value of the expression.

Given $\displaystyle{\cos{{\left({120}+{45}\right)}}}$

$\displaystyle{\cos{{\left({u}+{v}\right)}}}={\cos{{\left({u}\right)}}}{\cos{{\left({v}\right)}}}-{\sin{{\left({u}\right)}}}{\sin{{\left({v}\right)}}}$

$\displaystyle{\cos{{\left({120}+{45}\right)}}}={\cos{{\left({120}\right)}}}{\cos{{\left({45}\right)}}}-{\sin{{\left({120}\right)}}}{\sin{{\left({45}\right)}}}$

$\displaystyle{\cos{{\left({120}+{45}\right)}}}={\left(-\frac{{1}}{{2}}\right)}{\left(\frac{\sqrt{{2}}}{{2}}\right)}-{\left(\frac{\sqrt{{3}}}{{2}}\right)}{\left(\frac{\sqrt{{2}}}{{2}}\right)}=-\frac{\sqrt{{2}}}{{4}}-\frac{\sqrt{{6}}}{{4}}=\frac{{-\sqrt{{2}}-\sqrt{{6}}}}{{4}}$

Given $\displaystyle{\cos{{\left({120}\right)}}}+{\cos{{\left({45}\right)}}}$

$\displaystyle{\cos{{\left({120}\right)}}}+{\cos{{\left({45}\right)}}}={\left(-\frac{{1}}{{2}}\right)}+\frac{\sqrt{{2}}}{{2}}=\frac{{-{1}+\sqrt{{2}}}}{{2}}$

Given $\displaystyle{\cos{{\left({120}+{45}\right)}}}$

$\displaystyle{\cos{{\left({u}+{v}\right)}}}={\cos{{\left({u}\right)}}}{\cos{{\left({v}\right)}}}-{\sin{{\left({u}\right)}}}{\sin{{\left({v}\right)}}}$

$\displaystyle{\cos{{\left({120}+{45}\right)}}}={\cos{{\left({120}\right)}}}{\cos{{\left({45}\right)}}}-{\sin{{\left({120}\right)}}}{\sin{{\left({45}\right)}}}$

Given $\displaystyle{\cos{{\left({120}\right)}}}+{\cos{{\left({45}\right)}}}$

$\displaystyle{\cos{{\left({120}\right)}}}+{\cos{{\left({45}\right)}}}={\left(-\frac{{1}}{{2}}\right)}+\frac{\sqrt{{2}}}{{2}}=\frac{{-{1}+\sqrt{{2}}}}{{2}}$