If sin θ + cos θ = 1,

According to the question,

sin θ + cos θ = 1

As, sin θ + cos θ = 1

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

We know that,

$\sin (\pi / 4)=\cos (\pi / 4)=1 / \sqrt{2}$

$\Rightarrow \sqrt{2}\left(\sin \theta \cos \frac{\pi}{4}+\sin \frac{\pi}{4} \cos \theta\right)=1$

We know that

$\sin (A+B)=\sin A \cos B+\cos A \sin B$

$\Rightarrow \sin \left(\frac{\pi}{4}+\theta\right)=\frac{1}{\sqrt{2}}$

$\Rightarrow \sin \left(\frac{\pi}{4}+\theta\right)=\sin \frac{\pi}{4}$

Since we know,

If sin θ = sinα ⇒ θ = nπ + (-1)nα

We get,

θ + π/4 = nπ + (-1)n(π/4)

⇒ θ = nπ + (π/4)((-1)n – 1)