Did I Make a Mistake in Analyzing the Oscillating String?

[LCSphysicist]

Homework Statement

That's an easy problem. I will post below.

Relevant Equations

All below.

I thought we could do the part a by two steps:

(Studying the motion along x {there is no})

and along y

(Here should be T + dT)******

But apparently the answer is: "

"
So i think if the answer is wrong or, if not, where i made a mistake?

 

[hutchphd]

They are the same aren't they?

 

[LCSphysicist]

They are the same aren't they?

Jesus TRUTH, just now i notice the notation, omg zzzzzzzzzzzzz