Question

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

Answer 1

This can be simplified to `(sinxcosy + cosxsiny)/(cosxsinx - sinycosy)`

Explanation:

Rewrite `tan(x + y)` as `sin(x + y)/cos(x + y)`.

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

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

We now expand using the formulae `sin(A + B) = sinAcosB + cosAsinB`, `sin(A - B) = sinAcosB - cosAsinB` and `cos(A + B) = cosAcosB - sinAsinB`.

`=(sinxcosy + cosxsiny)/((cosxcosy - sinxsiny)(sinxcosy - cosxsiny)`

`=(sinxcosy + cosxsiny)/((cosxsinxcos^2y - sin^2xsinycosy + cosxsinxsin^2y - cos^2xcosysiny)`

Rearrange in order to look for a factorization in the denominator:

`=(sinxcosy + cosxsiny)/((cosxsinxcos^2y + cosxsinxsin^2y - sin^2xsinycosy - cos^2xcosysiny)`

`=(sinxcosy + cosxsiny)/(cosxsinx(cos^2y + sin^2y) - sinycosy(sin^2x + cos^2x))`

Recall that `sin^2theta + cos^2theta = 1`:

`=(sinxcosy + cosxsiny)/(cosxsinx - sinycosy)`

This is as far as we can go.

Hopefully this helps!