Derivatives of trigonometric functions

Homework Statement

I'm learning now about darivative all by my self (without a teacher) and I'm not sure about this development

The Attempt at a Solution

$$Y=senx$$

$$y+{\Delta}Y=sen(x+{\Delta}Y)$$

$${\Delta}Y=senx*cos{\Delta}x+ sen{\Delta}x*cosx$$

What about this cos,what is it doing in the equation?
I really don't get it.

this comes from the double angle formula.

it says that: sin (A + B) = sinAcosB + sinBcosA

you can find the same thing for sin (x + dY)

Oh,now I think i get it!
Thankyou!