<s>[INST] The maximum value of a trigonometric expression (without derivative) For how trivial it may appear, here I am with this question: find the value of $x$ for which this expression has its maximum.
$$f(x) = 1 + 4\sin^2(x) + 4\sin(x)\cos(x)$$
Without making use of derivatives, and only basing on trigonometric expressions / formulas. 
What I have tried is to manipulate a bit the expression, but I couldn't be able to reach any interesting expression.
 [/INST] Using 
$2 \, \sin^2 x = 1-\cos(2x)$, 
$\sin(2x)=2\,\sin x \, \cos x$, 
and $\sin(\theta)-\cos(\theta) = \sqrt 2 \, \sin (\theta - \pi/4)$,
we get:
\begin{align}
f(x) 
& = 1 + 4\sin^2(x) + 4\sin(x)\cos(x) \\ 
& = 1 + 2 - 2\cos(2x) + 2\sin(2x) \\
& = 3 + 2(\sin(2x)-\cos(2x)) \\
& = 3 + 2 \,\sqrt 2 \, \sin (2x-\pi/4) 
\end{align}
So the max of $f(x)$ should be $3+2\sqrt 2$.
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