Some help with proving trig identities and other things

I need some help with proving some trig identities step-by-step, justifying each step with either the word 'algebra' or the name of the fundamental trig identity used, and changing only one side to match the other (no adding one to both sides, for example)

The first is

(1 + csc x)/ sec x = cos x + cot x,

which I can see will involve cos x = 1/sec x somehow, but I can't see how to make cot x into 1 + csc x.

The next is

tan(x - π/4) = tan x - 1/1 + tan x,

which I can see may involve the cofunction identities, though again I don't see how.

The last is

cos x/sec x - 1 - cos x/ sec x + 1 = 2cot^2 x * cos x

I've simplified the left side to 2cosx/sec^2x - 1, which may or may not be helpful.

The next set of problems I'm having difficulty with is finding all solutions to both of the following equations the equations on the interval [0, 2pi).

I simply have no idea how to do these- what's the domain/range (whichever's applicable) for these functions again? Does it even matter?

2 cos x + sin 2x = 0

cot(x/2) = 1 - cos x/1 + cos x

Help me out!

Toptomcat