so, when verifying trig identities, a general rule is that you only want to manipulate one side of your equation. usually if you mess with both sides, you can get a little lost with the equation. the problems are created so that you only have to use one side.
cot²x - cot²x cos²x = cos²x
the first thing you can do here is factor a cot²x out of the left side of your equation:
cot²x (1 - cos²x) = cos²x
look inside the parentheses. you should recognize that as something you can make out of one of the pythagorean trig identities -- sin²x + cos²x = 1
to get this identity to match the inside of your parentheses, subtract cos²x:
sin²x = 1 - cos²x
so now you can substitute sin²x into your parentheses:
cot²x (1 - cos²x) = cos²x
cot²x (sin²x) = cos²x
recall that cot is the inverse of tan; tangent can be represented by (sin x/cos x) and cot is the reverse of that: (cos x/sin x)
(cos²x/sin²x)(sin²x) = cos²x
your sin²x's cancel out, as one is being divided and the other is being multiplied, and you're left with:
cos²x = cos²x
