C2 Trig Identities

When solving equations, I sometimes miss out solutions because I've manipulated the equation incorrectly. How can I make sure I don't do this?
For example:
$\tan 2\theta=5\sin 2\theta$
I could divide by tan throughout, but this would miss solutions out.
I know I need to multiply by cos, but how do I know which is the correct thing to do in general for questions like this?

When solving equations, I sometimes miss out solutions because I've manipulated the equation incorrectly. How can I make sure I don't do this?
For example:
$\tan 2\theta=5\sin 2\theta$
I could divide by tan throughout, but this would miss solutions out.
I know I need to multiply by cos, but how do I know which is the correct thing to do in general for questions like this?

When solving equations, I sometimes miss out solutions because I've manipulated the equation incorrectly. How can I make sure I don't do this?
For example:
$\tan 2\theta=5\sin 2\theta$
I could divide by tan throughout, but this would miss solutions out.
I know I need to multiply by cos, but how do I know which is the correct thing to do in general for questions like this?

Just try and make it so that you are manipulating only 1 trig function. Not really sure why you would divide by tan anyway.

Slightly curious, what would you get if you divided through by tan?

Wouldn't you end up with 5cos(2θ)=1?
Since tan(x) = sin(x)/cos(x)

No, that is the issue, hence why I asked how the OP would solve x^2 = x

I'd solve x^2-x by factorising into x(x-1), but I don't see what you're getting at.
I would have divided by tan to get 5cos(2θ)=1, but I know it misses out solutions.

The two are closely related. You didin't divide both sides by a function with the algebraic equation I gave yet did divide the trig one by a function (tan2θ)
By writing tan as sin/cos you will keep the two functions and be able to factor them accordingly.

Draw tan(2θ) and 5sin(2θ) on the same axis and see the number points of intersection (as with y = x^2 and y = x)