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