Quick questions on Differentiation of trigonometric functions (TCE Mathematics Methods, Tasmania)

For a function of the form $\sin(f(x))$, the chain rule multiplies by the derivative of the inside:

To locate stationary points you still solve $f'(x) = 0$, but now the solutions are angles and there are infinitely many of them. For $y = \sin x + \cos x$, $y' = \cos x - \sin x = 0$ gives $\tan x = 1$, so $x = \dfrac{\pi}{4} + n\pi$ for integer $n$. Within a stated domain such as $0 \le x \le 2\pi$ you list only the solutions that fall inside, here $x = \dfrac{\pi}{4}$ (a maximum) and $x = \dfrac{5\pi}{4}$ (a minimum). Always respect the given domain and remember that periodic functions repeat their stationary points.