Question

How do you simplify `sin(u-v)cosv+cos(u-v)sinv`?

Answer 1

`sinu`

Explanation:

`"using the "color(blue)"addition formulae"`

`•color(white)(x)sin(x+-y)=sinxcosy+-cosxsiny`

`•color(white)(x)cos(x+-y)=cosxcosy∓sinxsiny`

`•color(white)(x)sin^2x+cos^2x=1`

`sin(u-v)cosvlarrcolor(red)"first term"`

`=(sinucosv-cosusinv)cosv`

`=sinucos^2vcancel(-cosucosvsinv)`

`cos(u-v)sinvlarrcolor(red)"second term"`

`=(cosucosv+sinusinv)sinv`

`=cancel(cosucosvsinv)+sinusin^2v`

`"adding the 2 expansions gives"`

`sinucos^2v+sinusin^2v`

`=sinu(cos^2v+sin^2v)=sinu`