Solve the equation: $\sin 3x=2\cos^3x$ Solve the equation  :
$$\sin 3x=2\cos^3x$$
my try :
$\sin 3x=3\sin x-4\sin^3x$
$\cos^2x=1-\sin^2x$
so:
$$3\sin x-4\sin^3x=2((1-\sin^2x)(\cos x))$$
then ?
 A: Hint:
$$\sin 3x=2\cos^3x$$
$$3\sin x-4\sin^3x=2\cos^3x$$
$$3\sin x(\cos^2x+\sin^2x)-4\sin^3x=2\cos^3x$$
$$\sin^3x-3\sin x\cos^2x+2\cos^3x=0$$
Divide both sides by $\cos^3x$ and $\tan x=t$
$$t^3-3t+2=0$$
$$(t-1)^2(t+2)=0$$
A: $$2\cos^3x=3\sin x-4\sin^3x$$
Divide both sides by $\cos^3x$
$$2=3\tan x(1+\tan^2x)-4\tan^3x$$
$$\tan^3x-3\tan x+2=0$$ which is a cubic equation in $\tan x$
Clearly, one of the roots is $\tan x=1$
A: 1st of all you need to learn the formula Sin 3x and remember it so that whenever you see such a question you will choose the right way.
1st learn it . This may help you to learn this
https://youtu.be/He4JXYBwTj4
Come to our answer
sin 3x = 2 cos³ x
3sinx-4sin³x=2cos³x
(here look at the expression should you expand it in terms of cubic terms, obviously no it will mess up as in your case, it will be like doing so much thing for getting a small result. So here we will rather than expanding it we will divide cos³x on rhs. )
3tanxsec²x-4tan³x=2
Let tan x = t
So,
3 t (1+t²)-4t³=2
Then solve it and get
Tan x = 1 and -2
Hope it helps.
