Derivative of trigonometric functions?

[tbhaxor]

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} ?

[tbhaxor]

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

[TMosh]

@tbhaxor

Shouldn’t it be it \cos{x} - \cos{(x + \Delta x)}cosx−cos(x+Δx)\cos{x} - \cos{(x + \Delta x)} which is \cos{x} - \cos{x}\cos{\Delta x} + \sin{x} \sin{\Delta x}cosx−cosxcosΔx+sinxsinΔx\cos{x} - \cos{x}\cos{\Delta x} + \sin{x} \sin{\Delta x} ?

Yes.

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

[TMosh]

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

Proof of the derivative of sin(x) (video) | Khan Academy