How do you prove #[sin(x+ y) - sin(x-y)] /[ cos(x+ y) + cos(x-y)]= tan y#?

1 Answer
Jun 26, 2016

see explanation

Explanation:

To #color(blue)"Prove"# we require to manipulate one side into the same form as the other side.This will involve using #color(blue)"Addition formulae"#

#color(orange)"Reminders"#

#color(red)(|bar(ul(color(white)(a/a)color(black)(sin(A±B)=sinAcosB±cosAsinB)color(white)(a/a)|)))#
#color(red)(|bar(ul(color(white)(a/a)color(black)(cos(A±B)=cosAcosB∓sinAsinB)color(white)(a/a)|)))#

Starting with the left side and simplifying numerator/denominator separately.

Numerator

#sinxcosy+cosxsiny-[sinxcosy-cosxsiny)#

#=cancel(sinxcosy)+cosxsiny-cancel(sinxcosy)+cosxsiny#

#=2cosxsiny#

Denominator

#cosxcosy-sinxsiny+cosxcosy+sinxsiny#

#=cosxcosy-cancel(sinxsiny)+cosxcosy+cancel(sinxsiny)#

#=2cosxcosy#
#"---------------------------------------------------------------"#

left side can now be expressed as

#(2cosxsiny)/(2cosxcosy)=(cancel(2)cancel(cosx)siny)/(cancel(2)cancel(cosx)cosy)=(siny)/(cosy)#

and #(siny)/(cosy)=tany="right side hence proved"#