The Context Panel options Expand and Combine≻trig give the following solution.

$\sin \left(x\+y\right)$$\stackrel{\text{expand}}{\=}$$\sin \left(x\right)\cos \left(y\right)\+\cos \left(x\right)\sin \left(y\right)$$\stackrel{\text{combine}}{\=}$$\sin \left(x\+y\right)$

$\cos \left(x-y\right)$$\stackrel{\text{expand}}{\=}$$\cos \left(x\right)\cos \left(y\right)\+\sin \left(x\right)\sin \left(y\right)$$\stackrel{\text{combine}}{\=}$$\cos \left(x-y\right)$

$\sin \left(x\+y\right)$

$\mathrm{expand}\left(\sin \left(x\+y\right)\right)$

$\mathrm{combine}\left(\right)$

$\cos \left(x-y\right)$

$\mathrm{expand}\left(\cos \left(x-y\right)\right)$

$\mathrm{combine}\left(\cos \left(x-y\right)\right)$

$\mathrm{simplify}\left(q\,\left\{{\sin \left(x\right)}^2\+{\cos \left(x\right)}^2-1\right\}\,\left[\cos \left(x\right)\right]\right)$