Prove the identity? cos(x-y)sinx-sin(x-y)cosx=siny

• Identify the problem statement.
• Correctly use appropriate identities and/or theorems.
• Correctly use the algebraic process.
• Identify the final statement.

1 Answer
Apr 19, 2018

See below

Explanation:

#cos(x-y)sinx-sin(x-y)cosx=siny#

Cosine difference identity:
#(cosxcosy+sinxsiny)sinx-sin(x-y)cosx=siny#

Sine difference identity:
#(cosxcosy+sinxsiny)sinx-(sinxcosy-cosxsiny)cosx=siny#

Simplify:
#(cosxcosysinx+sin^2xsiny)-(sinxcosycosx-cos^2xsiny)=siny#

#cancel(cosxcosysinx)+sin^2xsinycancel(-sinxcosycosx)+cos^2xsiny=siny#

#sin^2xsiny+cos^2xsiny= siny#

Factor out #siny#:
#siny(sin^2x+cos^2x)= siny#

Apply the Pythagorean identity: #sin^2x+cos^2x=1#
#siny(1)= siny#

#siny=siny#