<s>[INST] Finding all solutions to $ \tan^5x - 9\tan{x} = 0 $ I am stuck when it comes to finding the end value of a trig function. I have the following question:
$$ \tan^5x - 9\tan{x} = 0 $$
I worked the problem and got:
$$ \tan x = 0\\
\tan^4x-9 = 0\\
x = 0, \pi, \frac {\pi}{3}, \frac {2\pi}{3}, \frac {4\pi}{3}, \frac {5\pi}{3} $$
My book answer is $x = \frac {\pi k}{3}$ how do you get that? I understand that tan uses $ \pi $ and sin, cos use $ 2\pi $ but I'm not sure how they got to that answer.
 [/INST] $$\tan^4x=9\Longrightarrow \tan^2x= 3\Longrightarrow \tan x=\pm\sqrt 3=\pm\frac{\frac{\sqrt 3}{2}}{\frac{1}{2}}=\pm\frac{\sin\frac{\pi}{3}}{\cos\frac{\pi}{3}}$$
Can you see it now?
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