In the figure on the left, the area of the large rectangle is S=(cosβ+cosα)(sinβ+sinα).

= sinβcosβ+cosβsinα+cosαsinα+sinβcosα+sinαcosα= sin(α+β)+sinβcosβ+sinαcosα。

The area of a big rectangle minus the area of four right-angled triangles is equal to the area of the shaded part S 1.

The area of the blank part is equal to the area of four right triangles, that is, 2× (12sinβ cosβ+12sinα cosα) = sinβ cosβ+sinα cosα.

So the area of the shadow is s1= s-sin β cos β+sin α cos α = sin (α+β).

And the area S2 of the shaded part in the right picture? Is equal to the sum of the areas of two small shadow rectangles, namely S2 = sin α cos β+cos α sin β.

In the picture on the right, the area of the big rectangle is also equal to S, and S2 is equal to the area of the big rectangle minus the areas of two small blank rectangles.

And the sum of the areas of two blank rectangles, namely sinβcosβ+sinαcosα,

Therefore, the area of the blank in the left picture is equal to the area of the blank in the right picture.

Therefore, the areas of the shaded parts in the left and right diagrams are also equal, that is, S 1 =S2, so there is sin(α+β)=sinαcosβ+cosαsinβ.

So the answer is: sin (α+β) = sin α cos β+cos α sin β.