Question: Find the minimum of $\\ f(x)=\frac{5\cos(x)-2\sin²(x)+4\sin(x)-3}{6|\cos(x)|+1}$ Attempt. So of course I tried calculus by differentiating but the derivative was overly complicated and couldn't manage to solve it, also I'm wondering if there is a non-calculus method. Which is what I tried right after, by considering the inequalities of the range of $\cos$ and $\sin$, of $-1≤\sin(x)≤1$ but only could manage to do it with trial and error:
Edit for those who asked for the trial and error method: You want the denominator to be smaller so that the outcome is bigger, and the numerator to be negative to find the minimum. So you start with the denominator which you want to be small as possible, hence by $|\cos(x)|=0$ which gives $x=-\frac{π}{2}$ which then you find out that in the numerator it outputs a negative number which pretty much concludes that is the least number you could get, which was $-9.$ And I checked on desmos and it was right.
But it isn't elegant. So I was wondering on how to do it.
