Question

How do you simplify `sin(x+y)+tan(x-y)*cos(x+y)` to trigonometric functions of x and y?

Answer 1

Use the following identities:

[1] `" " tan(x) = sin(x) / cos(x)`

[2] `" " sin(x+y) = sin(x)*cos(y) + cos(x)*sin(y)`

[3] `" " sin(x-y) = sin(x)*cos(y) - cos(x)*sin(y)`

[4] `" " cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y)`

[5] `" " cos(x-y) = cos(x)*cos(y) + sin(x)*sin(y)`

[6] `" " sin^2(x) + cos^2(x) = 1`

Thus, you can transform:

`sin(x+y) + tan(x - y) * cos(x + y)`

`stackrel("[1] ")(=) " " sin(x+y) + sin(x-y) / cos(x-y) * cos(x+y)`

`= " " sin(x+y) + (sin(x-y) cos(x+y))/cos(x-y)`

`stackrel("[2],[3],[4],[5]")(=) sin(x) cos(y) + cos(x) sin(y) + ((sin(x)cos(y) - cos(x) sin(y))*(cos(x)cos(y) - sin(x)sin(y))) / (cos(x)cos(y) + sin(x)sin(y))`

`= " " sin(x) cos(y) + cos(x) sin(y) + (sin(x)cos(x)cos^2(y) - sin(y)cos(y)cos^2(x) - sin(y)cos(y)sin^2(x) + sin (x)cos(x)sin^2(y)) / (cos(x)cos(y) + sin(x)sin(y))`

`= " " sin(x) cos(y) + cos(x) sin(y) + (color(green)(sin(x)cos(x)cos^2(y)) - sin(y)cos(y)cos^2(x) - sin(y)cos(y)sin^2(x) + color(green)(sin (x)cos(x)sin^2(y))) / (cos(x)cos(y) + sin(x)sin(y))`

`= " " sin(x) cos(y) + cos(x) sin(y) + ( sin(x)cos(x)(cos^2(y) + sin^2(y)) - sin(y)cos(y)(cos^2(x) + sin^2(x))) / (cos(x)cos(y) + sin(x)sin(y))`

`stackrel("[6] ")(=)" " sin(x) cos(y) + cos(x) sin(y) + (sin(x)cos(x) - sin(y)cos(y))/(cos(x)cos(y) + sin(x)sin(y))`

Unfortunately, I don't see how you could efficiently simplify this expression any further.

Hope that helped!