How do you simplify #tan(x+y)/sin(x-y)# to trigonometric functions of x and y?
1 Answer
This can be simplified to
Explanation:
Rewrite
#= ((sin(x + y))/cos(x + y))/(sin(x - y))#
#= sin(x + y)/(cos(x + y)sin(x - y)#
We now expand using the formulae
#=(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
#=(sinxcosy + cosxsiny)/(cosxsinx - sinycosy)#
This is as far as we can go.
Hopefully this helps!