cos(2x)-1=2cos²(x)-2=2(cos(x)-1)(cos(x)+1).
This expansion contains the expression in the denominator so for the purposes of finding the limit we can divide the fraction top and bottom by cos(x)-1 which leaves us with 2(cos(x)+1). When we substitute x=0 for the limit we get 4, so the limit is 4.
There is a calculator trick you can use which can tell you the limit quickly. In this case you put x equal to a value very close to the limit, say x=0.0001, and use the calculator to work out the fraction. It won’t matter whether x is in degrees or radians for this case. On my calculator, working in degrees I get 4.0046... which is obviously close to 4, confirming the earlier result derived algebraically. If I work in radians, I get the answer 4. This technique is useful as a check.