Double check these proofs using trig IDs

[rhombuster]

Sorry to post again but I really struggle with calculus. I don't know how to make formulas on this site so I'm posting it as a picture. I need someone to double check if these trig rules and algebra were applied correctly. Also unsure of -0/2 being possible. Thanks in advance!

 

[Ishuda]

For the second way I would do
\(\displaystyle 1 - cos(x) = 1 - cos[2 (\frac{x}{2})] = 2 sin^2(\frac{x}{2})\)
so
\(\displaystyle \lim_{x\to0}\frac{1 - cos(x)}{x} = \lim_{x\to0}\frac{2 sin^2(\frac{x}{2})}{x} = ...\)

Other than that, things appear correct to me.