Answer
$$\Delta y = \sin x\cos \Delta x + \sin \Delta x\cos x - \sin x\,\,\,{\text{ }}\,{\text{and}}\,\,\,{\text{ }}dy = \cos xdx$$
Work Step by Step
$$\eqalign{
& {\text{Let }}y = \sin x \cr
& \cr
& {\text{Calculating }}\Delta y.\,\,\,f\left( x \right) = \sin x \cr
& \Delta y = f\left( {x + \Delta x} \right) - f\left( x \right) \cr
& {\text{Then evaluating }} \cr
& \Delta y = \sin \left( {x + \Delta x} \right) - \sin x \cr
& {\text{Use sin}}\left( {A + B} \right) = \sin A\cos B + \sin B\cos A \cr
& \Delta y = \sin x\cos \Delta x + \sin \Delta x\cos x - \sin x \cr
& \cr
& {\text{and}} \cr
& y = \sin x \cr
& \frac{{dy}}{{dx}} = \cos x,{\text{ so }}dy = \cos xdx \cr
& \cr
& \Delta y = \sin x\cos \Delta x + \sin \Delta x\cos x - \sin x\,\,\,{\text{ }}\,{\text{and}}\,\,\,{\text{ }}dy = \cos xdx \cr} $$
