Explanation:

Using Pythagorean Identity

\(\displaystyle{{\sin}^{{2}}{x}}+{{\cos}^{{2}}{x}}={1},\ \text{ so }\ {{\cos}^{{2}}{x}}={1}-{{\sin}^{{2}}{x}}\)

\(\displaystyle{\cos{{x}}}=\pm\sqrt{{{1}-{{\sin}^{{2}}{x}}}}\)

\(\displaystyle{\sin{{x}}}+{\cos{{x}}}={\sin{{x}}}\pm\sqrt{{{1}-{{\sin}^{{2}}{x}}}}\)

Using complement / cofunction identity

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

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

Using Pythagorean Identity

\(\displaystyle{{\sin}^{{2}}{x}}+{{\cos}^{{2}}{x}}={1},\ \text{ so }\ {{\cos}^{{2}}{x}}={1}-{{\sin}^{{2}}{x}}\)

\(\displaystyle{\cos{{x}}}=\pm\sqrt{{{1}-{{\sin}^{{2}}{x}}}}\)

\(\displaystyle{\sin{{x}}}+{\cos{{x}}}={\sin{{x}}}\pm\sqrt{{{1}-{{\sin}^{{2}}{x}}}}\)

Using complement / cofunction identity

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

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