TITLE: Solving $\cos x+\sin x-1=0$
QUESTION [7 upvotes]: How does one solve this equation?

$$\cos {x}+\sin {x}-1=0$$

I have no idea how to start it.
Can anyone give me some hints? Is there an identity for $\cos{x}+\sin{x}$?
Thanks in advance!

REPLY [3 votes]: I'll throw my hat in the ring to get a picture in edgewise. :-)

I'll use $\theta$ for the angle, rather than $x$ (to avoid confusion with the $x$-coordinate).  If we then let $x = \cos \theta, y = \sin \theta$, then the allowable results are along the red circle with $x^2+y^2 = 1$ (since $\cos^2 \theta + \sin^2 \theta = 1$).  The solutions of the given equation are at the intersections of the blue line $x+y = 1$ with that red circle, yielding $(\cos \theta, \sin \theta) = (1, 0)$ and $(0, 1)$.
This in turn yields
$$
\theta = 2k\pi, \qquad \theta = 2k\pi + \frac{\pi}{2}
$$
for $k$ an integer.