Solve the equation sin⁡x+cos⁡x=ksin⁡xcos⁡x for real x, where k is a real constant.

The case of ${\displaystyle k=0}$ is elementary. Then squaring both members, 
${\displaystyle 2\cos x\sin x+1=k^2{\left(\cos x\sin x\right)}^2}$ 
immediately gives 
${\displaystyle \cos x\sin x=\frac{1\pm \sqrt{k^2+1}}{k^2}}$ 
and ${\displaystyle \cos x+\sin x=\frac{1\pm \sqrt{k^2+1}}{k}}$. 
This is a classical sum/product problem, solved with 
$\cos x-\sin x=\pm \sqrt{(\cos x+\sin x{)}^2-4\cos x\sin x} =\pm \frac{\sqrt{(1\pm \sqrt{k^2+1}{)}^2-4(1\pm \sqrt{k^2+1})}}{k}$