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Implicit Differentiation help

I found this question on the following site:

http://archives.math.utk.edu/visual.calculus/3/implicit.4/

It is the third from the bottom and looks like this:

y(tan(x+y)) = 4

I understand the entire solution to this, right until the last step. From my understanding, the simplification involves a trigonometric identity, and possibly some rearranging of the original formula. Any attempts I've made to move towards their final result have ended up in making the problem more complicated. Any hints anyone?

PS: I finshed school in June having done maths a year early and taken the FSMQ (top grades for both) and I knew I preferred calculus from the topics within FSMQ, so I began to study it more in depth. This isn't a calculus issue within the simplification (to my knowledge), but instead a trig identity, those of which I haven't yet studied.

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