TITLE: How to solve $\sin x +\cos x = 1$?
QUESTION [5 upvotes]: No matter how I do it, I always end up with $x = 0, 90, 270$ and $360$. All of those except $270$ is right, but I can't quite figure out how to get the $270$ degrees out of the answer. I've tried using trig identities, I've tried squaring both sides, but I always end up with $$2\sin x\cos x$$ which then leads me to $x = 0, 90, 270, 360$. But $$\sin (270) + \cos (270) = -1$$ so I'm doing something wrong.

REPLY [2 votes]: Using Double-Angle Formulas,
$$\sin x+\cos x=1\implies 2\sin\frac x2\cos\frac x2=1-\cos x=2\sin^2\frac x2$$
$$\implies \sin\frac x2\left(\cos\frac x2-\sin\frac x2\right)=0$$
$$(i)\sin\frac x2=0\implies \frac x2=n\pi\text{ where }n \text{ is any integer}$$
$$(ii) \cos\frac x2-\sin\frac x2=0\implies\cos\frac x2=\sin\frac x2\iff \tan\frac x2=1$$ Find the rest in my other answer