Chain Rule Differentiation: Simplifying Trigonometric Expressions

[grace77]

The question:

This is the solution that was given by my teacher

Attempt:

I understand how the work is done until the 3-4 line. Where did the 1-cos2x disappear to in the 4th line?
I know you can use the outside inside method but try as I might, I can't seem to understand how the final answer was gotten??

Can someone please tell me what I'm missing here??

 

[D H]

What is $(1+\cos(2x)) + (1-\cos(2x))$ ?

 

[Quesadilla]

The differentiation is done from line 1 to line 2. The rest is just tidying things up a little. The equality from line 3 to line 4 follows simply because

\begin{equation*}
2\sin(2x)(1 + \cos(2x)) + 2\sin(2x)(1 - \cos(2x)) =2 \sin(2x)( 1 + \cos(2x) + 1 - \cos(2x)) = 4 \sin(2x).
\end{equation*}