sin x + cos x =1 [0,360 deg)
$\displaystyle A \sin x + B \cos x$ can always be written in the form $\displaystyle C \sin (x + \phi)$.
So expand $\displaystyle C \sin (x + \phi)$ and compare it with $\displaystyle A \sin x + B \cos x$ to get equations that you can use to solve for $\displaystyle C$ and $\displaystyle \phi$.
An alternative approach to solving the original equation is given here: Solving Equations Involving Cosine Plus Sine
Note that when using this approach you must always test the resulting solutions to see whether or not they satisfy the original equation.