Answer:
Hence, all the solutions in the interval [0,2π) are:
\(0\ ,\pi\ ,\dfrac{\pi}{3}\ ,\dfrac{2\pi}{3}\ ,\dfrac{4\pi}{3}\ ,\dfrac{5\pi}{3}\)
Explanation:
We are asked to find the solution of the trignometric identity which is given by:
\(7\tan^3x-21\tan x=0\)
On dividing both side by 7 we get:
\(\tan^3x-3\tanx=0\\\\i.e.\\\\\tan x(\tan^2 x-3)=0\)
i.e.
Either
\(\tan x=0\)
i.e.
\(x=0,\pi\)
or
\(\tan^2x-3=0\\\\i.e.\\\\\tan^2x=3\\\\i.e.\\\\\tan x=\pm \sqrt{3}\)
If
\(\tan x=\sqrt{3}\\\\Then\\\\x=\dfrac{\pi}{3},\dfrac{4\pi}{3}\)
and if
\(\tan x=-\sqrt{3}\\\\Then\\\\x=\pi-\dfrac{\pi}{3}=\dfrac{2\pi}{3}\\\\and\\\\x=2\pi-\dfrac{\pi}{3}\\\\i.e.\\\\x=\dfrac{5\pi}{3}\)
Hence, all solutions are:
\(0\ ,\pi\ ,\dfrac{\pi}{3}\ ,\dfrac{2\pi}{3}\ ,\dfrac{4\pi}{3}\ ,\dfrac{5\pi}{3}\)
