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#