Question

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

Answer 1

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"`