I tried following the answers but yh this question is something. Thanks in advance
What's the confusion? Is there a particular part about the answers that makes no sense? (Sorry, but your question is generic and I don't know where to focus my response on.)
Line 1: Cosine double angle formula: cos(2x) = 2cos
2(x) - 1 so cos(x) = 2cos
2(x/2) - 1
Line 1 second part: Expand out the compound angle
Line 2: As with line 1 albeit with minus not plus
Line 3: Sums the above. This makes sense - we are just deriving something similar to product to sum. The inspiration to use squares and not to the power of 1 is simply because the result to prove involves squares.
Line 4: They tell you what they did. This isn't exactly classic in the HSC (classic for me) but you should anticipate it. Because how else do you turn an equation into an inequality
Line 5: Rearrange
_________
(ii) is using the inequality in (i) over and over again. This is not uncommon: The proof of the AM-GM inequality for 4 variables is just the AM-GM for 2 variables using the same method.
I.e. (x+y)/2 ≤ √(xy) and (a+b+c+d)/4 ≤
4√(abcd)
_________
There's hints in (iii) to use part (ii). What was perhaps not obvious was that 1/3(a+b+c) also had to be there. But look at it closely - you proved something involving multiplying 4 different cosines. You had to bring it out somehow.
Their typeface is pretty tedious to read though; I'll give that.