What am I missing in this Trigonometric Identity Simplification?

[jddoxtator]

Wanted to thank you guys once again.
I have done more complex problems and a lot of them, and I am now able to simplify trigonometric identities in 4 to 5 lines instead of whole pages of circular logic, reliably.

 

[mario99]

Wanted to thank you guys once again.
I have done more complex problems and a lot of them, and I am now able to simplify trigonometric identities in 4 to 5 lines instead of whole pages of circular logic, reliably.

You are welcome. You will see the power of these trigonometric identities later when you study Integration.

Look at this thread, post #13 to have a taste of what I am talking about!

ambigus question

here is the question A circle has its center at the point (0,1) is placed there stationary. Then it was given a push and it moved one full revolution without slipping. An observer traced the point (0,0) on the circle and noticed that it made a cycloid during the full revolution. What is the...

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-Mario, Dan's student

 

[Steven G]

Going from AB + AC to A ( B + C ) is surely something that you learned in Algebra. I bet that a significant amount of time was spent on that. It is called factoring.
I recall once when I was having trouble with an advanced math problem when my teacher asked me to state a particular theorem (a theorem is like AB + AC = A ( B + C )). I stated the theorem correctly, even knew the proof of the theorem but did not (yet) have the mathematical maturity to use it when it was needed--which was for the problem that I was having trouble with.
This is happening to you and if you keep doing problems and asking for help you will know when to use what you already learned.
You certainly are on the correct path. Good for you.

 

[jddoxtator]

So I cam across another one with a step that I have no clue how to do.
On 23 in the photo,I see how they get every step accept for the step from the second last to the last step.
I have never come across this.
Why would 2 (cos(x) sin(x)) + cos^2(x) + sin^2(x) = (sin(x) + cos(x))^2?
I can't think of any theorem or identity that allows for this.
Simplifying doesn't work and none of the base identity arrangements seem to work.

Edit: Never mind, I am clearly tired. The second to last step rearranges into the expansion of the square of the final answer.