If cos θ + sin θ = 2–√ cos θ,

Given that if $\cos \theta+\sin \theta=\sqrt{2} \cos \theta$, then we have to prove that $\cos \theta-\sin \theta=\sqrt{2} \sin \theta$

We have,

$\cos \theta+\sin \theta=\sqrt{2} \cos \theta$

$\Rightarrow \quad \sin \theta=\sqrt{2} \cos \theta-\cos \theta$

$\Rightarrow \quad \sin \theta=(\sqrt{2}-1) \cos \theta$

$\Rightarrow \quad \sin \theta=\frac{(\sqrt{2}-1)(\sqrt{2}+1)}{\sqrt{2}} \cos \theta$

$\Rightarrow \quad \sin \theta=\frac{\cos \theta}{\sqrt{2}+1}$

$\Rightarrow \quad \cos \theta=\sqrt{2} \sin \theta+\sin \theta$

$\Rightarrow \cos \theta-\sin \theta=\sqrt{2} \sin \theta$

Hence proved.