Maths question help - trigonometry

Given that (sin θ + cos θ ) ≠ 0, find all the solutions of
[2 cos 2θ(sin 2θ - √3 cos 2θ)] / (sin θ + cos θ) = √6 (sin 2 θ – √3 cos 2 θ )
for 0 ≤ θ < 360°.

Please help, I'm not getting too far with this!
Thank you

Given that (sin θ + cos θ ) ≠ 0, find all the solutions of
[2 cos 2θ(sin 2θ - √3 cos 2θ)] / (sin θ + cos θ) = √6 (sin 2 θ – √3 cos 2 θ )
for 0 ≤ θ < 360°.

Please help, I'm not getting too far with this!
Thank you

Given that (sin θ + cos θ ) ≠ 0, find all the solutions of
[2 cos 2θ(sin 2θ - √3 cos 2θ)] / (sin θ + cos θ) = √6 (sin 2 θ – √3 cos 2 θ )
for 0 ≤ θ < 360°.

Please help, I'm not getting too far with this!
Thank you

firstly divide the bracket (sin 2 θ – √3 cos 2 θ ) and get solutions by R transformation

the remaining equation on tidy up will produce a difference of squares (cosθ + sinθ)(cosθ - sinθ)

can you see what to do next?

What a horrible looking equation - where did this come from?

OK, to start with note that both sides have a common factor sin 2 θ – √3 cos 2 θ so EITHER this is 0 OR 2 cos 2θ / (sin θ + cos θ) = √6 (by dividing out that common factor).

Can you make some progress from here?

I can find tan2θ = √3
but I'm confused on the other solutions

It is AEA
I can do tan2θ = √3 , but I'm still confused on how I get solutions from 2 cos 2θ / (sin θ + cos θ) = √6 :-/

Do as TeeEm suggested:

multiply both sides by sin θ + cos θ first.

Then write out the standard identity for cos 2θ and use the fact that this is a difference of two squares.

Note the common factor on both sides of your new equation and the fact you are given that a certain expression is not zero so you can divide by it.

Solve what is left

To solve it, do I need to use Rcos(θ+α) form?

If you've ended up with something like cos A - sin A = something

then yes - you can probably virtually write down what R and alpha are going to be

Ah it's not working any help?

What equation have you ended up with and how are you trying to solve it?

I'm a bit tired to be trying this sort of thing now, so if someone doesn't help you see it through tonight I'll be back on tomorrow, but you're probably only a step or two away from getting to the end now

What equation have you ended up with and how are you trying to solve it?

I'm a bit tired to be trying this sort of thing now, so if someone doesn't help you see it through tonight I'll be back on tomorrow, but you're probably only a step or two away from getting to the end now

I am not sure why there is any harmonic form going on?

I've only been skimming this online instead of writing it out properly, but it looks like it's going to end up with solving something like

cos A - sin A = something

at the end unless I've missed something in the working.

So it's about 1 step away from being able to write down the solution.

I've got to θ+α = 30
but I think I've missed somewhere what α is

It's fine if you can't do it now I'm sure I'll figure it out, thanks for your help!

You basically want to be thinking about cos(X + 30) or some multiple of it (X = theta cos I'm lazy)

Okay, thank you I think I can have a mess around with this last bit now, and hopefully come to the answers