How do you verify the identity $sin(x-y)=(1-cotxtany)/(cscxsecy)$ ?

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Answer 1 · 2016-12-18T19:30:24

See proof below

We use

$sin(A-B)=sinAcosB-sinBcosA$

$cotx=cosx/sinx$

$tany=siny/cosy$

$cscx=1/sinx$

$secy=1/cosy$

The LHS is

$sin(x-y)=sinxcosy-cosxsiny$

The RHS is

$(1-cotxtany)/(cscxsecy)$

$=(1-cosx/sinx*siny/cosy)/(1/sinx*1/cosy)$

$=(sinxcosy-cosxsiny)/(cancelsinxcancelcosy*1/(cancelsinxcancelcosy))$

$=sinxcosy-cosxsiny=sin(x-y)$

$=LHS$

$Q. E. D$

How do you verify the identity sin(x-y)=(1-cotxtany)/(cscxsecy)sin(x-y)=(1-cotxtany)/(cscxsecy)?