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: $180, 195, 120, 135, 360$ 
 A: \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 $\boxed{195^{\circ}}$.
A: Here's a plot that should help:

A: Do some trigonometry and find the general solution first:
$$\tan^23x=2\sin^23x=\frac{2\tan^23x}{1+\tan^23x}\iff\tan^23x(1+\tan^23x)=2\tan^23x$$
So,


*

*either $\tan3x=0\iff 3x\equiv 0\mod180\iff x\equiv 0\mod 60$;

*or $\tan^23x=1\iff\tan 3x=\pm 1\iff 3x\equiv \pm45\mod 180\iff x\equiv\pm 15\mod 60$.


Then select the values in $\bigl[0,90]$.
A: If $\cos6x=a,$
Using $\cos2y=1-2\sin^2y=\dfrac{1-\tan^2y}{1+\tan^2y}$
$$\dfrac{1-a}{1+a}=1-a$$
$$\implies a(a-1)=0$$
If $\cos6x=0,6x=(2n+1)90^\circ$ where $n$ is an integer
If $\cos6x=1,6x=360^\circ m$
