<s>[INST] what is $x$ for $\tan^23x = 2\sin^23x$ If $x$ = acute angle then find $x$ such that $\tan^23x = 2\sin^23x$.
So 
$\tan3x = \sqrt2\sin3x$
$\frac{1}{\cos3x} = \sqrt2$
$3x = 45^{\circ}$
what are all the possibilies for $x$ ?
because the question asked for all possibilities of $x$
The options are: $180, 195, 120, 135, 360$ 
 [/INST] \begin{align}
\tan^23x&=2\sin^23x\\
\tan^23x-2\sin^23x&=0\\
\sin^23x(\frac1{\cos^23x}-2)&=0\\
\sin^23x(\sec^23x-2)&=0
\end{align}
So we have either
\begin{align}
\sin^23x&=0\\
3x&=0^{\circ}, 180^{\circ}, 360^{\circ}\\
x&=0^{\circ}, 60^{\circ}, 120^{\circ}
\end{align}
or
\begin{align}
\sec^23x-2&=0\\
\sec^23x&=2\\
\sec3x&=\pm\sqrt2\\
3x&=45^{\circ}, 135^{\circ}, 225^{\circ}, 315^{\circ}\\
x&=15^{\circ}, 45^{\circ}, 75^{\circ}, 105^{\circ}
\end{align}
From the above, the only admissible solutions are $\boxed{15^{\circ}, 45^{\circ}, 60^{\circ}, 75^{\circ}}$.
The sum of these angles is $\boxed{195^{\circ}}$.
</s>