\(\displaystyle{\left|{\sin{{x}}}\right|}+{\left|{\cos{{x}}}\right|}\geq{\left|{\sin{{x}}}\right|}^{{2}}+{\left|{\cos{{x}}}\right|}^{{2}}={1}\)

and for upper bound (remember \(\displaystyle{\left({a}+{b}\right)}^{{2}}\leq{2}{\left({a}^{{2}}+{b}^{{2}}\right)}\)):

\(\displaystyle{\left|{\sin{{x}}}\right|}+{\left|{\cos}\right|}{x}\leq\sqrt{{{2}{\left({\left|{\sin{{x}}}\right|}^{{2}}+{\left|{\cos{{x}}}\right|}^{{2}}\right)}}}=\sqrt{{2}}\)