2 csc 2x = sec x csc x

LHS = 2 csc 2x

= 2 / sin 2x

= 2 / 2 sin x cos x

= 1 / sin x cos x

= csc x sec x

= RHS

Thefore, proven.

2 cot 2x = cot x - tan x

LHS = 2 cot 2x

= 2 / tan 2x

= 2 / [2 tan x / 1 - tan^2 x]

= 2[1 -tan^2 x] / 2 tan x

= 1 - tan^2 x / tan x

= cot x -tan x

= RHS

Therefore, proven.

and ...

cos6x + sin6x = 1 - 3 sin2x + 3 sin4x

Let us check if that is true when x = 30 degrees,

cos(6*30deg) +sin(6*30deg) =? 1 -3sin(2*30deg) +3sin(4*30deg)

-1 +0 =? 1 -3(0.866) +3(0.866)

-1 =? 1

No.

So, cos6x + sin6x = 1 - 3 sin2x + 3 sin4x is not an identity.