Let's do it together.
We start with the point (,) in a coordinate plane. This point is presented in the polar form, 
and we want to learn what is a curve   = .

Do you know that  =  ?

It is the direct consequence of the addition formula for cosine.
Also, it is named the formula for double argument for cosine.
If you don't know it, then look in these lessons 

   Addition and subtraction formulas
   Trigonometric functions of multiply argument

OK. Having this, let's make one step further.

 =  = . = ..   (1)

Now,    is x-component of the point  (, )  in the rectangular coordinate system on the coordinate plane. 
So we can write  = .

Similarly,    is y-component of the point  (,)  in the rectangular coordinate system on the coordinate plane. 
So we can write  = .

Therefore, we can rewrite the formula (1) in this way:

 = . =  =  .     (3)

Then your original equation   =   becomes 
 = .

Do you recognize this equation ?  What is this ?  Is it familiar to you ?


But of course, it is the equation of a hyperbola . . . 


So, we unraveled this mystery.