<s>[INST] Find the exact value of trigonometric expression: $ \arctan\frac{2\cos \frac{\pi}{30} \cos\frac{\pi}{15}}{1+2\cos \frac{\pi}{30} \sin\frac{\pi}{15}}$? How can I simplify this trigonometric expression?

$$  \arctan\frac{2\cos \frac{\pi}{30} \cos\frac{\pi}{15}}{1+2\cos \frac{\pi}{30} \sin\frac{\pi}{15}}$$

I used 
$$\sin \frac{\pi}{15}=2 \sin \frac{\pi}{30}  \cos \frac{\pi}{30}$$ and $$\cos\frac{\pi}{15}=2\cos^2\frac{\pi}{30}-1$$
But these give me more complicated expressions.
 [/INST] The identity can also be proved from: $\quad\cos36° = \cos 72° + {1\over2}$ 
$\cos 72° + {1\over2}= (2\cos^2 36° - 1)+ {1\over2}\;= \large{\phi^2\over2}-{1\over2} ={\phi \over 2}=\cos36°$

$T= \Large \frac{2\cos 6° \cos 12°}{1\;+\;2\cos 6° \sin 12°}$ 
$\Large{1\over T} = {1\over 2\cos 6° \cos 12°} + \normalsize \tan 12°$
$\begin{align} \tan 36° - \tan12° 
&= {\sin36° \over \cos36°} - {\sin12° \over \cos12°} \cr
&= {\sin36° \cos12° - \cos36° \sin12° \over \cos12° \cos36°} \cr
&= {\sin24° \over \cos12° (\cos72° + 0.5)} \cr
&= {\sin24° \over \cos12°(\sin18° + \sin30°)} \cr
&= {\sin24° \over \cos12°(2 \sin24° \cos6°)} \cr
\tan36° &= {1\over 2\cos 6° \cos 12°} + \tan12°
\end{align}$
$$T = \cot 36° = \tan 54°$$
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