Question:Find the minimum vertical distance between the graphs of $2+\sin x$ and $\cos x$?
In order to find out the required distance, what should I do? It seems that there is a problem if I differentiate the equation $y = 2+\sin x - \cos x$ and solve $x$ when $y=0$. Would anyone mind telling me how to solve it?
Create your distance function $$f(x)=2+\sin x - \cos x$$ and take its derivative $$f'(x)=\cos x+\sin x.$$ The minima and maxima occur where $f'(x)=0$, i.e., $$\sin x=-\cos x.$$ There are trigonometry techniques for finding those points. The maxima occur where the second derivative is negative, and the minima are where the second derivative is positive (if the second derivative were zero it might not be either a maximum or minimum). $$f''(x)=-\sin x+\cos x.$$ You can easily see all this and check your work approximately by plotting the function.
If it isn't clear to you (after a little research) how you can find the solutions to the equation $\sin x=-\cos x$ above, I would suggest asking that as a separate question.