True
A counterexample is a special type of example that is an exception to a general rule, law, proposition or statement. Hence if we prove that [tex]sin(45^{\circ})+cos(45^{\circ})\neq 1[/tex] then this is a counterexample for the expression [tex]sin(x)+cos(x)=1[/tex]. Thus:
[tex]sin(x)+cos(x)=1 \\ \\ Substituting \ x=45^{\circ} \ we \ have: \\ \\ sin(45^{\circ})+cos(45^{\circ})=1 \\ \\\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}=1 \\ \\ Rewriting:\\ \\2\frac{\sqrt{2}}{2}=1 \\ \\ \therefore \sqrt{2}=1 \ FALSE![/tex]
So we observe that the equation is not fulfilled because [tex]\sqrt{2}\neq 1[/tex]. In conclusion:
A counterexample for the expression sin(x)+cos(x)=1 is 45° is true!
