We draw a right triangle with legs having the lengths of those
two coefficients.  We put the 20 on the adjacent side to y
and 21 on the opposite side, so that we can use the given 
sin(x+y) formula.  [If we did it the other way we'd have to 
use the cos(x-y) formula.]

 

Note here that y is the angle whose tangent is 21/20, and we
state that mathematically by the equation y = tan-1(21/20).

We calculate the hypotenuse by the Pythagorean theorem:








 

We multiply the given expression, which is 



by  which is the same as multiplying
by 1, and does not change the value:



We distribute the 



We use the right triangle above to substitute trig
ratios for the fractions:



We rearrange the factors in the parentheses so that 
what's inside the parentheses will look like the right 
side of the given formula sin(x+y)=sin(x)cos(y)+sin(y)cos(x)



We replace the parentheses using the given formula:

, where y = tan-1(21/20).

Edwin