Trig Identity Question Finding Value of K

Question

If $\sin(x) + \cos(x) = k$ for what value(s) of $k$ can $\sin(x)\cos(x)=1$?

Answer 1

hint $$(\cos x+\sin x)^2=k^2$$ $$(1+2\cos x \sin x)=k^2$$

Answer 2

Hint:

$$\sin(x)=k-\cos(x)$$

$$\to\cos(x)(k-\cos(x))=1$$ $$\to k\cos(x)-\cos^2(x)=1$$ $$\to\cos^2(x)-k\cos(x)+1=0$$ $$\chi=\cos(x)\implies \chi^2-k\chi+1=0$$