How does the base of this orange triangle was taken \Delta \cos{x}? Shouldn’t it be it \cos{x} - \cos{(x + \Delta x)} which is \cos{x} - \cos{x}\cos{\Delta x} + \sin{x} \sin{\Delta x} ?

Also why we are taking \phi on top, instead of opposite side?

Yes.

Maybe they’ve done some simplification or other algebra tricks. I haven’t done any trig in a long time.

Here is a video that shows the algebra for a similar derivative process:

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