<s>[INST] Prove this trigonometry inequality I'm having difficulty proving that tan(26°) < 0.5 < tan(27°) . Any idea ? Thanks. 
 [/INST] OK, here is the solution. Feel free to ask if you don't understand any of the steps.
Let's first prove the left side of the inequality:
$\tan26 < \frac{1}{2}  \Rightarrow 2sin26<cos26 \Rightarrow 4sin^2{26}<cos^2{26}$
$\Rightarrow 4sin^2{26} < 1-sin^2{26} \Rightarrow sin^2{26}<\frac{1}{5} \Rightarrow cos^2{52}>\frac{3}{5} \Rightarrow cos^2{52}>\frac{9}{25} $
$\Rightarrow cos104>\frac{-7}{25} \Rightarrow sin14< \frac{7}{25}$ ---    But sin14 < sin15
also $sin\frac{\pi}{12}<\frac{\pi}{12}<\frac{7}{25} \Rightarrow\Rightarrow tan26<\frac{1}{2}$   
Now let's prove the right side :
$tan27>\frac{1}{2} \Rightarrow 4sin^2{27}>cos^2{27} \Rightarrow sin^2{27}>\frac{1}{5}$
$\Rightarrow cos54<\frac{3}{5} \Rightarrow sin36<{3}{5} \Rightarrow sin^2{36}<\frac{9}{25} \Rightarrow cos72>\frac{7}{25} \Rightarrow sin18>\frac{7}{25}$
But  $sin18=\frac{\sqrt{5}-1}{4} > \frac{1.2}{4} > \frac{7}{25}$   
So $sin18>\frac{7}{25} \Rightarrow \Rightarrow tan27>\frac{1}{2}$
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